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THE THIRTEEN BOOKS 

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THE THIRTEEN BOOKS OF 
EUCLID'S ELEMENTS 

TRANSLATED FROM THE TEXT OF HEIBERG 

WITH INTRODUCTION AND COMMENTARY 



BY 

Sir THOMAS L. HEATH, 

K.C.B., K.C.V.O., F.R.S., 

SC.D. CAMS,, HON. D.SC. OXFORD 
HONORARY FELLOW (SOMETIME FELLOW) OF TRINITY COLLEGE CAMBRIDGE 

SECOND EDITION 
REVISED WITH ADDITIONS 



VOLUME I 
INTRODUCTION AND BOOKS I, H 



DOVER PUBLICATIONS, INC. 

NEW YORK 









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Published in the United Kingdom by Constable 
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WC 2. 



This Dover edition, first published in 1956, is an 
unabridged and unaltered republication of the 
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PREFACE 

" ' I 'HERE never has been, and till we see it we never 
J. shall believe that there can be, a system of geometry 
worthy of the name, which has any material departures (we do 
not speak of corrections or extensions or developments) from 
the plan laid down by Euclid." De Morgan wrote thus in 
October 1 848 {Short supplementary remarks on the first six 
Books of Euclid's Elements in the Companion to the Almanac 
for 1 849) ; and I do not think that, if he had been living 
to-day, he would have seen reason to revise the opinion so 
deliberately pronounced sixty years ago. It is true that in the 
interval much valuable work has been done on the continent 
in the investigation of the first principles, including the 
formulation and classification of axioms or postulates which 
are necessary to make good the deficiencies of Euclid's own 
explicit postulates and axioms and to justify the further 
assumptions which he tacitly makes in certain propositions, 
content apparently to let their truth be inferred from observa- 
tion of the figures as drawn ; but, once the first principles are 
disposed of, the body of doctrine contained in the recent text- 
books of elementary geometry does not, and from the nature 
of the case cannot, show any substantial differences from that 
set forth in the Elements. In England it would seem that far 
less of scientific value has been done ; the efforts of a multitude 
of writers have rather been directed towards producing alter- 
natives for Euclid which shall be more suitable, that is to say, 
easier, for schoolboys. It is of course not surprising that, in 



vi PREFACE 

these days of short cuts, there should have arisen a movement 
to get rid of Euclid and to substitute a "royal road to 
geometry " ; the marvel is that a book which was not written 
for schoolboys but for grown men (as all internal evidence 
shows, and in particular the essentially theoretical character 
of the work and its aloofness from anything of the nature of 
"practical" geometry) should have held its own as a school- 
book for so long. And now that Euclid's proofs and arrange- 
ment are no longer required from candidates at examinations 
there has been a rush of competitors anxious to be first in the 
field with a new text- book on the more "practical" lines which 
now find so much favour. The natural desire of each teacher 
who writes such a text -book is to give prominence to some 
special nostrum which he has found successful with pupils. 
One result is, too often, a loss of a due sense of proportion ; 
and, in any case, it is inevitable that there should be great 
diversity of treatment. It was with reference to such a danger 
that Lardner wrote in 1846 : "Euclid once superseded, every 
teacher would esteem his own work the best, and every school 
would have its own class book. All that rigour and exactitude 
which have so long excited the admiration of men of science 
would be at an end. These very words would lose all definite 
meaning. Every school would have a different standard; 
matter of assumption in one being matter of demonstration in 
another ; until, at length, Geometry, in the ancient sense of 
the word, would be altogether frittered away or be only 
considered as a particular application of Arithmetic and 
Algebra." It is, perhaps, too early yet to prophesy what will 
be the ultimate outcome of the new order of things ; but it 
would at least seem possible that history will repeat itself and 
that, when chaos has come again in geometrical teaching, 
there will be a return to Euclid more or less complete for the 
purpose of standardising it once more. 

But the case for a new edition of Euclid is independent of 
any controversies as to how geometry shall be taught to 
schoolboys. Euclid's work will live long after all the text-books 



PREFACE vii 

of the present day are superseded and forgotten. It is one 
of the noblest monuments of antiquity ; no mathematician 
worthy of the name can afford not to know Euclid, the real 
Euclid as distinct from any revised or rewritten versions 
which will serve for schoolboys or engineers. And, to know 
Euclid, it is necessary to know his language, and, so far as it 
can be traced, the history of the *' elements " which he 
collected in his immortal work. 

This brings me to the raison d'itre of the present edition. 
A new translation from the Greek was necessary for two 
reasons. First, though some time has elapsed since the 
appearance of Heiberg's definitive text and prolegomena, 
published between 1883 and 1888, there has not been, so far 
as I know, any attempt to make a faithful translation from it 
into English even of the Books which are commonly read. 
And, secondly, the other Books, vn. to X. and xin., were not 
included by Simson and the editors who followed him, or 
apparently in any English translation since Williamson's 
(178 1 — 8), so that they are now practically inaccessible to 
English readers in any form. 

In the matter of notes, the edition of the first six Books 
in Greek and Latin with notes by Camerer and Ha'uber 
(Berlin, 1824 — 5) is a perfect mine of information. It would 
have been practically impossible to make the notes more 
exhaustive at the time when they were written. But the 
researches of the last thirty or forty years into the history of 
mathematics (I need only mention such names as those of 
Bretschneider, Hankel, Moritz Cantor, Hultsch, Paul Tannery, 
Zeuthen, Loria, and Heiberg) have put the whole subject 
upon a different plane. I have endeavoured in this edition 
to take account of all the main results of these researches up 
to the present date. Thus, so far as the geometrical Books 
are concerned, my notes are intended to form a sort of 
dictionary of the history of elementary geometry, arranged 
according to subjects ; while the notes on the arithmetical 
Books vii. — ix. and on Book x. follow the same plan. 



viii PREFACE 

I desire to express here my thanks to my brother, 
Dr R. S. Heath, Vice- Principal of Birmingham University, 
for suggestions on the proof sheets and, in particular, for the 
reference to the parallelism between Euclid's definition of 
proportion and Dedekind's theory of irrationals, to Mr R. D. 
Hicks for advice on a number of difficult points of translation, 
to Professor A. A. Bevan for help in the transliteration of 
Arabic names, and to the Curators and Librarian of the 
Bodleian Library for permission to reproduce, as frontispiece, 
a page from the famous Bodleian MS, of the Elements. 
Lastly, my best acknowledgments are due to the Syndics of 
the Cambridge University Press for their ready acceptance 
of the work, and for the zealous and efficient cooperation of 
their staff which has much lightened the labour of seeing the 
book through the Press. 

T. L H. 



Novem$er t 1908. 



























PREFACE TO THE SECOND EDITION 

I LIKE to think that the exhaustion of the first edition of 
this work furnishes a new proof (if such were needed) 
that Euclid is far from being defunct or even dormant, and 
that, so long as mathematics is studied, mathematicians will 
find it necessary and worth while to come back again and 
again, for one purpose or another, to the twenty-two-centuries- 
old book which, notwithstanding its imperfections, remains the 
greatest elementary textbook in mathematics that the world is 
privileged to possess. 

The present edition has been carefully revised throughout, 
and a number of passages (sometimes whole pages) have been 
rewritten, with a view to bringing it up to date. Some not in- 
considerable additions have also been made, especially in the 
Excursuses to Volume I, which will, I hope, find interested 
readers. 

Since the date of the first edition little has happened in the 
domain of geometrical teaching which needs to be chronicled. 
Two distinct movements however call for notice. 

The first is a movement having for its object the mitigation 
of the difficulties (affecting in different ways students, teachers 
and examiners) which are found to arise from the multiplicity 
of the different textbooks and varying systems now in use for 
the teaching of elementary geometry. These difficulties have 
evoked a widespread desire among teachers for the establish- 
ment of an agreed sequence to be generally adopted in teaching 
the subject. One proposal to this end has already been made: 
but the chance of the acceptance of an agreed sequence has in 
the meantime been prejudiced by a second movement which 
has arisen in other quarters. 



x PREFACE TO THE SECOND EDITION 

I refer to the movement in favour of reviving, in a modified 
form, the proposal made by Wallis in 1663 to replace Euclid's 
Parallel- Postulate by a Postulate of Similarity (as to which see 
pp. 2 10 — 1 1 of Volume I of this work). The form of Postulate 
now suggested is an assumption that "Given one triangle, 
there can be constructed, on any arbitrary base, another triangle 
equiangular with (or similar to) the given triangle." It may 
perhaps be held that this assumption has the advantage of not 
referring, in the statement of it, to the fact that a straight line 
is of unlimited length ; but, on the other hand, as is well known, 
Saccheri showed ( 1 733) that it involves more than is necessary 
to enable Euclid's Postulate to be proved. In any case it 
would seem certain that a scheme based upon the proposed 
Postulate, if made scientifically sound, must be more difficult 
than the procedure now generally followed. This being so, 
and having regard to the facts ( 1 ) that the difference between 
the suggested Postulate and that of Euclid is in effect so slight 
and (2) that the historic interest of Euclid's Postulate is so 
great, I am of opinion that the proposal is very much to be 
deprecated. 

T. L. H. 

December 1925. 



CONTENTS 



VOLUME I. 



INTRODUCTION. 

Facsimile ofapageofthe Bodleian ms. of the Elements . Frontispiece 

This is a facsimile of a page (fol. 45 verso) of the famous Bodleian Ms. of 
the Element i, D'Orville 301 {formerly x. r inf. 1, 30), written in the year 
888. The scholium in the margin, not very difficult to decipher, though 
some letters are almost rubbed out, is one of the scholia Vatican a given by 
Heiberg (Vol. v. p. 163} as ill. No- 15 : Aid tw xirrpov tOcQv q&k rj v 
f»rn$«tft aftav, tl Six* Tifiwauti* dWiJXaT * t4 y&p icfwrpor a&r&r 1j Sixrrofdn, 
bjLoltits xai i} et r$F tripe* Bt& rov Kfrrpov aCenjt 17 trip* ^ 5ta rav KfvTpov tlTj, 
Sri 06 3fva r4fit*erai 1j && tou Ktvrpoy. The rt before U in the last sentence 
should be omitted, PFVat. read ij without el. The marginal references 
lower down are of course to propositions quoted, (1) Sti t& a' rod y. " by 



III. i," and (aj &i ri 7' toC atow, u by 3 of the same/ 1 



Chap. 



I. 

II. 
III. 
IV. 
V. 
VI. 
VII. 
VIII. 
IX. 



Euclid and the traditions about him 

Euclid's other works 

Greek commentators other than Proclus 
Proclus and his sources .... 

The Text 

The Scholia 

Euclid in Arabia ...... 

Principal translations and editions . 
§ 1. On the nature of Elements 
g a. Elements anterior to Euclid's . 
§ 3. First principles : Definitions, Postulates 
and Axioms ..... 

Theorems and Problems 

The formal divisions of a proposition 

Other technical terms 

The definitions 



§4- 
§5- 

§7- 



THE ELEMENTS. 

Definitions, Postulates, Common Notions 

Notes on Definitions etc. . 

Propositions .... 

Definitions .... 

Note on geometrical algebra 

Propositions .... 

Pythagoras and the Pythagoreans 
II. Popular names for Euclidean Propositions 

Greek Index to Vol. I 

English Index to Vol. I 



Book I. 



Book II. 



Excursus 
Excursus 



pack 
1 
7 

19 
*9 
46 
64 
75 
9i 
114 
116 

117 
124 
129 
132 
143 



1 S3 

i55 
241 

370 
37» 
375 
411 

415 

419 

421 



INTRODUCTION. 



CHAPTER I. 

EUCLID AND THE TRADITIONS ABOUT HIM. 

As in the case of the other great mathematicians of Greece, so in 
Euclid's case, we have only the most meagre particulars of the life 
and personality of the man. 

Most of what we have is contained in the passage of Proclus' 
summary relating to him, which is as follows 1 : 

"Not much younger than these (sc. Hermotimus of Colophon and 
Philippus of Med ma) is Euclid, who put together the Elements, collect- 
ing many of Eudoxus' theorems, perfecting many of Theaetetus', and 
also bringing to irrefragable demonstration the things which were 
only somewhat loosely proved by his predecessors. This man lived* 
in the time of the first Ptolemy. For Archimedes, who came imme- 
diately after the first (Ptolemy)', makes mention of Euclid: and, 
further, they say that Ptolemy once asked him if there was in 
geometry any shorter way than that of the elements, and he answered 
that there was no royal road to geometry*. He is then younger than 
the pupils of Plato but older than Eratosthenes and Archimedes ; for 
the latter were contemporary with one another, as Eratosthenes some- 
where says." 

This passage shows that even Proclus had no direct knowledge 
of Euclid's birthplace or of the date of his birth or death. He pro- 
ceeds by inference. Since Archimedes lived just after the first 

1 Proclus, ed. Fried lein, p. 68, 6 — io, 

* The word ytyott must apparently mean " nourished," as Heiberg understands it 
(LitttrttrgathicktHeht Stttditn itber EtMid, 188,1, p. 16), not "was born," as Hankel took 
it ; otherwise part of Proclus' argument would lose its cogency. 

* So Heiberg understands ?n/iii\£iv n? *p&Ti# (sc. IlToXe^ttV). Friedlein's text has 
vol between Iwifi&Xuv And t<£ *p&ry \ and it is right to remark that another reading is 
ttX it Tif Trptirif (without trtpaXiiy'i which has been translated " in his first book," by which 
is understood On the Sphere and Cylinder I., where (i) in Prop- 3 »™ the words " let BC 
be made equal to D by tkt second (proposition) of the ftrit of Euclid's (books)," and (i) in 
Prop, fi the words " For these things are handed down in the Elements " (without the name 
of Euclid). Heiberg thinks the former passage is referred to, and that Proclus must 
therefore have had before him the words " by the second of the first of Euclid ": a fair proof 
that they are genuine, though in themselves they would be somewhat suspicious. 

4 The same story is told in Stobaeus, Eel. (it. p. m8, 30, ed, Wachsmuthf about 
Alexander and Menaechmus. Alexander is represented as having asked Menaechmus to 
teach him geometry concisely, but he replied 1 "O king, through the country there are royal 
roads and road* for common citizen*, but in geometry there is one road for all." 



* INTRODUCTION [ch. i 

Ptolemy, and Archimedes mentions Euclid, while there is an anecdote 
about some Ptolemy and Euclid, therefore Euclid lived in the time of 
the first Ptolemy. 

We may infer then from Proclus that Euclid was intermediate 
between the first pupils of Plato and Archimedes. Now Plato died in 
347/6, Archimedes lived 287-2 1 2, Eratosthenes c. 284-204 B.C. Thus 
Euclid must have flourished c. 300 b.c, which date agrees well with 
the fact that Ptolemy reigned from 306 to 283 B.C. 

It is most probable that Euclid received his mathematical training 
in Athens from the pupils of Plato; for most of the geometers who 
could have taught him were of that school, and it was in Athens that 
the older writers of elements, and the other mathematicians on whose 
works Euclid's Elements depend, had lived and taught. He may 
himself have been a Platonist, but this does not follow from the state- 
ments of Proclus on the subject, Proclus says namely that he was of 
the school of Plato and in close touch with that philosophy 1 . But 
this was only an attempt of a New Platonist to connect Euclid with 
his philosophy, as is clear from the next words in the same sentence, 
" for which reason also he set before himself, as the end of the whole 
Elements, the construction of the so-called Platonic figures." It is 
evident that it was only an idea of Proclus' own to infer that Euclid 
was a Platonist because his Elements end with the investigation of 
the five regular solids, since a later passage shows him hard put to 
it to reconcile the view that the construction of the five regular solids 
was the end and aim of the Elements with the obvious fact that they 
were intended to supply a foundation for the study of geometry in 
general, "to make perfect the understanding of the learner in regard 
to the whole of geometry*." To get out of the difficulty he says* that, 
if one should ask him what was the aim {o-kowos) of the treatise, he 
would reply by making a distinction between Euclid's intentions 

( 1) as regards the subjects with which his investigations are concerned, 

(2) as regards the learner, and would say as regards ( 1 ) that " the 
whole of the geometer's argument is concerned with the cosmic 
figures." This latter statement is obviously incorrect It is true 
that Euclid's Elements end with the construction of the five regular 
solids ; but the pi an i metrical portion has no direct relation to them, 
and the arithmetical no relation at all ; the propositions about them 
are merely the conclusion of the stereo metrical division of the work. 

One thing is however certain, namely that Euclid taught, and 
founded a school, at Alexandria. This is clear from the remark of 
Pappus about Apollonius*: "he spent a very long time with the 
pupils of Euclid at Alexandria, and it was thus that he acquired 
such a scientific habit of thought," 

It is in the same passage that Pappus makes a remark which 
might, to an unwary reader, seem to throw some light on the 

1 Proclus, p. 68, JO, ul tJ rpoatpim ti lUaTwiKii itrn cal r^i 0i\<wo0Ja twJtj eJituw. 

1 i&jd, p. 7 [, 8. * ibid. p. 70, 19 jqq. 

* Pappus, VII, p, 678, 10 — II, CVffxokAjrat rat i>*& EvxXd&ov jukthjrait tr 'AX*(aj>?pr£p 



ch. i] EUCLID AND TRADITIONS ABOUT HIM 3 

personality of Euclid. He is speaking about Apollonius' preface 
to the first book of his Conies, where he says that Euclid had not 
completely worked out the synthesis of the " three- and four-line 
locus," which in fact was not possible without some theorems first 
discovered by himself. Pappus says on this 1 : "Now Euclid — 
regarding Aristaeus as deserving credit for the discoveries he had 
already made in conies, and without anticipating him or wishing to 
construct anew the same system (such was his scrupulous fairness and 
his exemplary kindliness towards all who could advance mathematical 
science to however small an extent), being moreover in no wise con- 
tentious and, though exact, yet no braggart like the other [Apollonius] 
— wrote so much about the locus as was possible by means of the 
conies of Aristaeus, without claiming completeness for his demonstra- 
tions." It is however evident, when the passage is examined in its 
context, that Pappus is not following any tradition in giving this 
account of Euclid : he was offended by the terms of Apollonius' 
reference to Euclid, which seemed to him unjust, and he drew a 
fancy picture of Euclid in order to show Apollonius in a relatively 
unfavourable light. 

Another story is told of Euclid which one would like to believe true. 
According to Stobaeus*, "some one who had begun to read geometry 
with Euclid, when he had learnt the first theorem, asked Euclid, ' But 
what shall I get bylearning these things?' Euclid called his slave 
and said 'Give him threepence, since he must make gain out of what 
he learns.' " 

In the middle ages most translators and editors spoke of Euclid 
as Euclid of Megara. This description arose out of a confusion 
between our Euclid and the philosopher Euclid of Megara who lived 
about 400 B.C. The first trace of this confusion appears in Valerius 
Maximus (in the time of Tiberius) who says* that Plato, on being 
appealed to for a solution of the problem of doubling the cubical 
aitar, sent the inquirers to " Euclid the geometer." There is no doubt 
about the reading, although an early commentator on Valerius 
Maximus wanted to correct " Eucliden " into " Eudoxum? and this 
correction is clearly right. But, if Valerius Maximus took Euclid the 
geometer for a contemporary of Plato, it could only be through 
confusing him with Euclid of Megara. The first specific reference to 
Euclid as Euclid of Megara belongs to the 14th century, occurring in 
the v^rofivtifuiTitrfial of Theodorus Metochita (d. 1332) who speaks of 
" Euclid of Megara, the Socratic philosopher, contemporary of Plato," 
as the author of treatises on plane and solid geometry, data, optics 
etc. : and a Paris MS. of the 14th century has " Euctidis philosophi 
Socratici liber ele mentor urn," The misunderstanding was general 
in the period from Campanus 1 translation (Venice 1482) to those of 
Tartaglia (Venice 1565) and Candalla (Paris 1566). But one 
Constantinus Lascaris (d. about 1493) had already made the proper 

1 Pappus, VII. pp. €76, 55 — 678* 6. Hultsch, it is true, brackets the whole passage 
ppu 676, i j— 678, I j, but apparently on the ground of the diction only, 
* Stobaeus, I.e. * vill. n, eit. i. 



4 INTRODUCTION [ch. i 

distinction by saying of our Euclid that "he was different from him 
of Megara of whom Laertius wrote, and who wrote dialogues "' ; and 
to Commandinus belongs the credit of being the first translator* to 
put the matter beyond doubt : " Let us then free a number of people 
from the error by which they have been induced to believe that our 
Euclid is the same as the philosopher of Megara " etc 

Another idea, that Euclid was born at Gela in Sicily, is due to tne 
same confusion, being based on Diogenes Laertius* description' of the 
philosopher Euclid as being "of Megara, or, according to some, of 
Gela, as Alexander says in the AiaBoxai-" 

In view of the poverty of Greek tradition on the subject even as 
early as the time of Proclus (410-485 A.D.), we must necessarily take 
cum gram the apparently circumstantial accounts of Euclid given by 
Arabian authors ; and indeed the origin of their stories can be 
explained as the result (1) of the Arabian tendency to romance, and 
(2) of misunderstandings. 

We read* that " Euclid, son of Naucrates, grandson of Zenarchus", 
called the author of geometry, a philosopher of somewhat ancient 
date, a Greek by nationality domiciled at Damascus, born at Tyre, 
most learned in the science of geometry, published a most excellent 
and most useful work entitled the foundation or elements of geometry, 
a subject in which no more general treatise existed before among the 
Greeks : nay, there was no one even of later date who did not walk 
in his footsteps and frankly profess his doctrine. Hence also Greek, 
Roman and Arabian geometers not a few, who undertook the task 
of illustrating this work, published commentaries, scholia, and notes 
upon it, and made an abridgment of the work itself. For this reason 
the Greek philosophers used to post up on the doors of their schools 
the well-known notice : ' Let no one come to our school, who has not 
first learned the elements of Euclid.' " The details at the beginning 
of this extract cannot be derived from Greek sources, for even Proclus 
did not know anything about Euclid's father, while it was not the 
Greek habit to record the names of grandfathers, as the Arabians 
commonly did. Damascus and Tyre were no doubt brought in to 
gratify a desire which the Arabians always showed to connect famous 
Greeks in some way or other with the East. Thus Nasiraddin, the 
translator of the Elements, who was of Tus in Khurasan, actually 
makes Euclid out to have been " Thusinus " also*. The readiness of 
the Arabians to run away with an idea is illustrated by the last words 

1 Letter to Fernandus Acuna, printed in Maurolycus, Hitteria Sitiliat, fol. 11 r. (see 
Heiberg, EuMid-Studieriy pp. n — 3, 35). 

■ Preface to translation (Pisauri, 1571). 

* Diog. L. 11. 106, p. ;8 ed. Cobet. 

* Casiri, Bibliotheca Arabifo-Hispana Escuriaiensis, 1. p. 339, Casirt's source is al- 
Qifti (d. 1 148). trie author of the Ta'rttk ai-ffu&amd, a collection of biographies of phi- 
losophers, mathematicians, astronomers etc 

■ The Fi'Aris/ says "son of Naucrates, the son of Berenice f/) " (see Suter"s translation in 
Abhandiungtn tur Gach. d. Math. V], Heft, 1H91. p. 16). 

e The same predilection made the Arabs describe 'Pythagoras as a pupil of the wise 
Salomo, Hippftrcbus as the exponent of Chaldaean philosophy or as the Chaldaean, Archi- 
medes as an Egyptian etc. (Hij! Khalfa, Lexicon BiMiogrspkitvm, and Casiri). 



ch. i) EUCLID AND TRADITIONS ABOUT HIM 5 

of the extract. Everyone knows the story of Plato's inscription over 
the porch of the Academy : " let no one unversed in geometry enter 
my doors " ; the Arab turned geometry into Euclid 's geometry, and 
told the story of Greek philosophers in general and ''their Academies," 

Equally remarkable are the Arabian accounts of the relation of 
Euclid and Apollonius 1 . According to them the Elements were 
originally written, not by Euclid, but by a man whose name was 
Apollonius, a carpenter, who wrote the work in 1 5 books or sections'. 
In the course of time some of the work was lost and the rest became 
disarranged, so that one of the kings at Alexandria who desired to 
study geometry and to master this treatise in particular first questioned 
about it certain learned men who visited him and then sent for Euclid 
who was at that time famous as a geometer, and asked him to revise 
and complete the work and reduce it to order. Euclid then re- wrote 
h in 13 books which were thereafter known by his name. (According 
to another version Euclid composed the 1 3 books out of commentaries 
which he had published on two books of Apollonius on conies and 
out of introductory matter added to the doctrine of the five regular 
solids.) To the thirteen books were added two more books, the work 
of others (though some attribute these also to Euclid) which contain 
several things not mentioned by Apollonius. According to another 
version Hypsicles, a pupil of Euclid at Alexandria, offered to the 
king and published Books xiv. and xv., it being also stated that 
Hypsicles had " discovered " the books, by which it appears to be 
suggested that Hypsicles had edited them from materials left by Euclid. 

We observe here the correct statement that Books XIV. and xv. 
were not written by Euclid, but along with it the incorrect informa- 
tion that Hypsicles, the author of Book XIV., wrote Book XV. also. 

The whole of the fable about Apollonius having preceded Euclid 
and having written the Elements appears to have been evolved out of 
the preface to Book xiv. by Hypsicles, and in this way ; the Book 
must in early times have been attributed to Euclid, and the inference 
based upon this assumption was left uncorrected afterwards when it 
was recognised that Hypsicles was the author. The preface is worth 
quoting : 

"Basiiides of Tyre, O Protarchus, when he came to Alexandria 
and met my father, spent the greater part of his sojourn with him on 
account of their common interest in mathematics. And once, when 

1 The authorities for these statements quoted by Casiri and lliji Khalfa are al-Kindl '3 
tract de inttituto tibri Eutlidis (al-Kindl died about 873) and a commentary by Qidliide 
ar-Ruml (<[. about 1440) on a book called Askk&l ai-ttf sis (fundamental propositions) by 
Ashraf Shanuaddln aS'Samaru&ndi (c. 1176) consisting of elucidations of 3$ propositions 
■elected from the first books of Euclid. Naslraddin likewise says that Euclid cut out two of 
15 books of elements then existing and published the rest under his own name. According to 
Qadtzade the king heard that there was a celebrated geometer named Euclid at Tyrt\ Naslr- 
atldin says that he sent for Euclid of Tus. 

* So says the Fihrist, Suter {<>p- cit, p. 49) thinks that the author of the Fihrist did not 
suppose Apollonius of Ptrga to be the writer of the Ettmmts, as later Arabian authorities 
did, but that he disUnguished another Apollonius whom he calls " a carpenter." Suter's 
argument is based on the fact that the Fihrist s article on Apollonius (of Perga) says nothing 
of the Ebmttus, aiiJ that it gives the three great mathematicians, Euclid, Archimedes and 
Apollonius, in the correct chronological order. 



6 INTRODUCTION [ch. i 

examining the treatise written by Apollonius about the comparison 
between the dodecahedron and the icosahedron inscribed in the same 
sphere, (showing) what ratio they have to one another, they thought 
that Apollonius had not expounded this matter properly, and 
accordingly they emended the exposition, as I was able to learn 
from my father. And I myself, later, fell in with another book 
published by Apollonius, containing a demonstration relating to the 
subject, and I was greatly interested in the investigation of the 
problem. The book published by Apollonius is accessible to all— 
for it has a large circulation, having apparently been carefully written 
out later — but I decided to send you the comments which seem to 
me to be necessary, for you will through your proficiency in mathe- 
matics in general and in geometry in particular form an expert 
judgment on what I am about to say, and you will lend a kindly ear 
to my disquisition for the sake of your friendship to my father and 
your goodwil! to me." 

The idea that Apollonius preceded Euclid must evidently have 
been derived from the passage just quoted. It explains other things 
besides. Basil ides must have been confused with j3a<7t\ev$, and we 
have a probable explanation of the " Alexandrian king," and of the 
" learned men who visited " Alexandria. It is possible also that in 
the " Tyrian " of Hypsicles" preface we have the origin of the notion 
that Euclid was born in Tyre. These inferences argue, no doubt, 
very defective knowledge of Greek : but we could expect no better 
from those who took the Organon of Aristotle to be " instrumentum 
musicum pneumaticum," and who explained the name of Euclid, 
which they variously pronounced as U elides or /eludes, to be com- 
pounded of Ueli a key, and Dis a measure, or, as some say, geometry, 
so that [/elides is equivalent to the key of geometry '! 

Lastly the alternative version, given in brackets above, which says 
that Euclid made the Elements out of commentaries which he wrote 
on two books of Apollonius on conies and prolegomena added to the 
doctrine of the five solids, seems to have arisen, through a like 
confusion, out of a later passage 1 in Hypsicles' Book XIV. : " And this 
is expounded by Aristaeus in the book entitled 'Comparison of the five 
figures,' and by Apollonius in the second edition of his comparison of 
the dodecahedron with the icosahedron." The "doctrine of the five 
solids" in the Arabic must be the "Comparison of the five figures" 
in the passage of Hypsicles, for nowhere else have we any information 
about a work bearing this title, nor can the Arabians have had. The 
reference to the two books of Apollonius on conies will then be the 
result of mixing up the fact that Apollonius wrote a book on conies 
with the second edition of the other work mentioned by Hypsicles. 
We do not find elsewhere in Arabian authors any mention of a 
commentary by Euclid on Apollonius and Aristaeus: so that the 
story in the passage quoted is really no more than a variation of the 
fable that the Elements were the work of Apollonius. 

1 Heibtrg*5 Euclid, vol. v. p. 6. 



CHAPTER II. 

EUCLID'S OTHER WORKS. 

In giving a list of the Euclidean treatises other than the Elements, 
I shall be brief: for fuller accounts of them, or speculations with 
regard to thern, reference should be made to the standard histories of 
mathematics 1 . 

I will take first the works which are mentioned by Greek authors, 

i. The Pseudaria. 

I mention this first because Proclus refers to it in the general 
remarks in praise of the Elements which he gives immediately after 
the mention of Euclid in his summary. He says'; " But, inasmuch 
as many things, while appearing to rest on truth and to follow from 
scientific principles, really tend to lead one astray from the principles 
and deceive the more superficial minds, he has handed down methods 
for the discriminative understanding of these things as well, by the 
use of which methods we shall be able to give beginners in this study 
practice in the discovery of paralogisms, and to avoid being misled. 
This treatise, by which he puts this machinery in our hands, he 
entitled (the book) of Pseudaria, enumerating in order their various 
kinds, exercising our intelligence in each case by theorems of all 
sorts, setting the true side by side with the false, and combining 
the refutation of error with practical illustration. This book then is 
by way of cathartic and exercise, while the Elements contain the 
irrefragable and complete guide to the actual scientific investigation 
of the subjects of geometry." 

The book is considered to be irreparably lost. We may conclude 
however from the connexion of it with the Elements and the reference 
to its usefulness for beginners that it did not go outside the domain 
of elementary geometry'. 

1 See, for example, Lena, Le seiente esatte neW antica Grata, 1914, pp. 145- — 168; 
T. L. Heath, History of Greek Mathematics, 191 1, J. pp. \i\ — 446. Cf- Heiberg, Litttrar- 
gatkUhtliche Studim tibar Euklid, pp. 36 — 153; Euclidis opera omnia, ed. Heiberg and 
Menge, Vols. VI. — VLI1, 

■ Proclus, p. 70, i — 18. 

3 Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his 
commentary on Aristotle's Sophistiti Ehncki {fol. 25 b) : "Not only those {fheyx *) which do 
not start from the principles of the science under which the problem is classed... but also 
those which do start from the proper principles of the science but in some respect admit a 
paralogism, e.g. the Ptevobgraphemaia of Euclid." Tannery {Butt, ties jeiencei ma/A- et aifr. 
»• Serie, vi., 1881, [*" Partie, p. 147) conjectures that it may be from this treatise that the 
: commentator got his Information about the quadratures of the circle by Antiphon and 



8 INTRODUCTION [ch. n 

2. The Data. 

The Data (Se&oftiva) are included by Pappus in the Treasury of 
Analysis (x6tto<; avrikvo/ievoi;), and he describes their contents 1 . They 
are still concerned with elementary geometry, though forming part 
of the introduction to higher analysis. Their form is that of pro- 
positions proving that, if certain things in a figure are given (in 
magnitude, in species, etc.), something else is given. The subject- 
matter is much the same as that of the planimetrical books of the 
Elements, to which the Data are often supplementary. We shall see 
this later when we come to compare the propositions in the Elements 
which give us the means of solving the general quadratic equation 
with the corresponding propositions of the Data which give the 
solution. The Data may in fact be regarded as elementary exercises 
in analysis. 

It is not necessary to go more closely into the contents, as we 
have the full Greek text and the commentary by Marinus newly 
edited by Menge and therefore easily accessible 1 . 

3. The book On divisions {of figures). 

This work (irepl Statpetre&v fiiffklov) is mentioned by Proclus*. 
In one place he is speaking of the conception or definition (\6yo<t) 
of figure, and of the divisibility of a figure into others differing from 
it in kind ; and he adds : " For the circle is divisible into parts unlike 
in definition or notion (eivo/ioia rji Xtfyy), and so is each of the 
rectilineal figures ; this is in fact the business of the writer of the 
Elements in his Divisions, where he divides given figures, in one case 
into like figures, and in another into unlike*." "Like" and "unlike" 
here mean, not "similar" and "dissimilar" in the technical sense, but 
"like" or "unlike in definition or notion" (\6y<p): thus to divide a 
triangle into triangles would be to divide it into "like" figures, to 
divide a triangle into a triangle and a quadrilateral would be to 
divide it into "unlike'' figures. 

The treatise is lost in Greek but has been discovered in the 
Arabic. First John Dee discovered a treatise De divisionibus by one 
Muhammad Bagdadinus* and handed over a copy of it (in Latin) in 
1563 to Commandinus. who published it, in Dee's name and his own, 
in 1570 s , Dee did not himself translate the tract from the Arabic; he 

Brjrccm, to say nothing of the lunules of Hippocrates. I think however that there is is 
objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson'i 
quadrature did not start from the proper principles of geometry, but from some principles 
more general. 

1 Pappus, vn. p. 638. 

* Vol. vi. in the Teubner edition of Euclidis Optra omnia by Heiberg and Menge. A 
translation of the Data is also included in Simson*3 Euclid (though naturally his text left 
much to be desired). 

8 Proclus, p. 69, 4. * ibid. 144, Ji— 26. 

* Stetnschneider places him in the roth c. H. Suter (BiUiothtea Mathematics, 1v„ 1903, 
pp. «4, 17} identifies him with Abu (Bekr) Muh. b. 'Abdalbaq! al-Bagdadl, QadI (Judge) of 
Maristan (circa rt>70-i 14 1), to whom he also attributes the Likcrjudci (? judicis) super detimum 
Euciidis translated by Gherard of Cremona. 

' Dc atperficicrum divisionibui liber Mackomcto Bagdadino adscripUa, nunc primam 
/oannis Da Londitunsis tt Federici Ctsmmandini Urbinatti opera in lutein cditus, Pisauri, 
1570, afterwards included in Gregory's Euclid (Oxford, r7oj). 



ch. n] EUCLID'S OTHER WORKS 9 

found it in Latin in a ms, which was then in his own possession but 
was about 20 years afterwards stolen or destroyed in an attack by a 
mob on his house at Mortlake'. Dee, in his preface addressed to 
Commandinus, says nothing of his having translated the book, but 
only remarks that the very illegible ms. had caused him much trouble 
and (in a later passage) speaks of " the actual, very ancient, copy from 
which I wrote out..." (in ipso unde descripsi vetustissimo exemplari). 
The Latin translation of this tract from the Arabic was probably made 
by Gherard of Cremona (1 1 14- 1 1 87), among the list of whose numerous 
translations a " liber divisionum " occurs. The Arabic original cannot 
have been a direct translation from Euclid, and probably was not even 
a direct adaptation of it ; it contains mistakes and un mathematical 
expressions, and moreover does not contain the propositions about 
the division of a circle alluded to by Proclus. Hence it can scarcely 
have contained more than a fragment of Euclid's work. 

But Woepcke found in a MS. at Paris a treatise in Arabic on the 
division of figures, which he translated and published in 1851*. It is 
expressly attributed to Euclid in the MS. and corresponds to the 
description of it by Proclus. Generally speaking, the divisions are 
divisions into figures of the same kind as the original figures, e.g. of 
triangles into triangles ; but there are also divisions into " unlike " 
figures, e.g. that of a triangle by a straight line parallel to the base. 
The missing propositions about the division of a circle are also here: 
" to divide into two equal parts a given figure bounded by an arc 
of a circle and two straight lines including a given angle " and " to 
draw in a given circle two parallel straight lines cutting off a certain 
part of the circle." Unfortunately the proofs are given of only four 
propositions (including the two last mentioned) out of 36, because 
the Arabic translator found them too easy and omitted them. To 
illustrate the character of the problems dealt with I need only take 
one more example : " To cut off a certain fraction from a (parallel-) 
trapezium by a straight line which passes through a given point lying 
inside or outside the trapezium but so that a straight line can be 
drawn through it cutting both the parallel sides of the trapezium," 
The genuineness of the treatise edited by Woepcke is attested by the 
facts that the four proofs which remain are elegant and depend on 
propositions in the Elements, and that there is a lemma with a true 
Greek ring: "to apply to a straight line a rectangle equal to the 
rectangle contained by AB, AC and deficient by a square." Moreover 
the treatise is no fragment, but finishes with the words "end of the 
treatise," and is a well-ordered and compact whole. Hence we may 
safely conclude that Woepcke's is not only Euclid's own work but 
the whole of it. A restoration of the work, with proofs, was attempted 
by Ofterdinger 3 , who however does not give Woepcke's props. 30, 31, 
34. 35. 36- VVe have now a satisfactory restoration, with ample notes 

1 R. C. Archibald, Euclid's Boot: on the Division of Figures with a restoration based on 
Wotpehts text and on the Praetita geometriae of Leonardo Pisano^ Cambridge, 19(5, pp. 4 — 9. 

' Journal A natiqut, |8JI, p. 133 sqq. 

3 L. F. Ofterdinger, Beitrtige tur Wiederherslettung dcr Sehrifi dot Eu&lides iibrr die 
Tkeilmtg der Figurtn, Ulm, 1BJ3. 



io INTRODUCTION [ch. n 

and an introduction, by R. C. Archibald, who used for the purpose 
Woepcke's text and a section of Leonardo of Pisa's Practica geometriae 
(1220) 1 . 

4. The Porisms. 

It is not possible to give in this place any account of the con- 
troversies about the contents and significance of the three lost books 
of Porisms, or of the important attempts by Robert Simson and 
Chasles to restore the work. These may be said to form a whole 
literature, references to which will be found most abundantly given 
by Heiberg and Loria, the former of whom has treated the subject 
from the philological point of view, most exhaustively, while the 
latter, founding himself generally on Heiberg, has added useful 
details, from the mathematical side, relating to the attempted restora- 
tions, etc' It must suffice here to give an extract from the only 
original source of information about the nature and contents of the 
Porisms, namely Pappus*. In his general preface about the books 
composing the Treasury of Analysis (towot dvaXuofttvos) he says : 

"After the Tan gene ies (of Apollonius) come, in three books, the 
Porisms of Euclid, [in the view of many] a collection most ingeniously 
devised for the analysis of the more weighty problems, [and] although 
nature presents an unlimited number of such porisms' 1 , [they have 
added nothing to what was written originally by Euclid, except that 
some before my time have shown their want of taste by adding to a 
few (of the propositions) second proofs, each (proposition) admitting 
of a definite number of demonstrations, as we have shown, and 
Euclid having given one for each, namely that which is the most 
lucid. These porisms embody a theory subtle, natural, necessary, 
and of considerable generality, which is fascinating to those who can 
see and produce results]. 

" Now all the varieties of porisms belong, neither to theorems nor 
problems, but to a species occupying a sort of intermediate position 
[so that their enunciations can be formed like those of either theorems 
or problems], the result being that, of the great number of geometers, 
some regarded them as of the class of theorems, and others of pro- 
blems, looking only to the form of the proposition. But that the 
ancients knew better the difference between these three things is 
clear from the definitions. For they said that a theorem is that 
which is proposed with a view to the demonstration of the very 
thing proposed, a problem that which is thrown out with a view to 
the construction of the very thing proposed, and a porism that which 
is proposed with a view to the producing of the very thing proposed. 
[But this definition of the porism was changed by the more recent 
writers who could not produce everything, but used these elements 

1 There is a remarkable similarity between the propositions of Wot octet's text and those 
of Leonardo, suggesting that Leonardo may have had before him a translation (perhaps by 
Gherard of Cremona) of the Arabic tract. 

* Heiberg, liu&lid-Studiin, pp. 56 — 79, and Loria, op. at., pp. 1 S3 — jfij. 

' Pappus, ed. Hullsch, VII. pp. 648 — 660. I put in square brackets the words bracketed 
by Haltsch. 

' I adopt Heiberg 1 s reading of a comma here instead of a full stop. 



ch. it] EUCLID'S OTHER WORKS n 

and proved only the fact that that which is sought really exists, but 
did not produce it 1 and were accordingly confuted by the definition 
and the whole doctrine. They based their definition on an incidental 
characteristic, thus : A porism is that which falls short of a locus- 
theorem in respect of its hypothesis'. Of this kind of porisms loci 
are a species, and they abound in the Treasury of Analysis ; but 
this species has been collected, named and handed down separately 
from the porisms, because it is more widely diffused than the other 
species]. But it has further become characteristic of porisms that, 
owing to their complication, the enunciations are put in a contracted 
form, much being by usage left to be understood; so that many 
geometers understand them only in a partial way and are ignorant of 
the more essential features of their contents, 

"[Now to comprehend a number of propositions in one enunciation 
is by no means easy in these porisms, because Euclid himself has not 
in fact given many of each species, but chosen, for examples, one or a 
few out of a great multitude*. But at the beginning of the first book 
he has given some propositions, to the number of ten, of one species, 
namely that more fruitful species consisting of loci.] Consequently, 
finding that these admitted of being comprehended in one enunciation, 
we have set it out thus: 

If, in a system of four straight lines' which cut each other 
two and two, three points on one straight line be given while the 
rest except one lie on different straight lines given in position, 
the remaining point also will lie on a straight line given in 
position*. 

1 Heiberg points out that Props. 5—9 of Archimedes' treatise On Spirals are porisms in 
this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K- 
It is then possible, says Archimedes, to draw a straight line r> b F 

KffF t meeting the circumference in // and the tangent in F, 
such that 

FH;HJC^{*kBH):c, 
where c is the circumference of any circle. To prove this he 
assumes the following construction. E being any straight line 
greater than c, he says 1 let KG be parallel to DF, "and let 
the line GH equal to E be placed verging to the point B." 
Archimedes must of course nave known how to effect this 
construction, which requires conies. But that it is possible requires very little argument, for 
if we draw any straight line BHG meeting the circle in /if and KG in G, it is obvious that 
as G moves away from C, HG becomes greater and greater and may be made as great as we 
please. The " later writers " would no doubt have contented themselves with this considera- 
tion without actually tanstrtuting HG. 

1 As Heiberg says, this translation is made certain by a preceding passage of Pappus 
(p. 648, 1 — 3) where he compares two enunciations, the latter of which " falls short of the 
former in hypothesis but goes beyond it in requirement" E.g. the first enunciation requiring 
us, given three circles, to draw a circle touching all three, the second may require us, given 
only row circles (one less datum), to draw a circle touching them and ef a given sat (an 
extra requirement). 

* I translate Heiberg 's reading with a full stop here followed by r/wi ipxi U tiu*t [rpb* 

&PXV* ($t&Oneror) Hultsch] TOU TpwTOU /Sl^X/ou,,., 

1 The four straight lines are described in the text as (the sides) brriov 1 9 wapuwrtout i.e. 
aides of two' sotts of quadrilaterals which Simson tries to explain {see p. no of the Index 
Grtueitatis of Hultsch's edition of Pappus). 

* In other words (Chasles, p, 13; Loria, p. 356), if a triangle be so deformed that each of 
its sides turns about one of three points in a straight line, and two of its vertices lie on two 
straight lines given in position, the third vertex will also lie on a straight line. 




is INTRODUCTION [ch. n 

"This has only been enunciated of four straight lines, of which not 
more than two pass through the same point, but it is not known (to 
most people) that it is true of any assigned number of straight lines 
if enunciated thus : 

If any number of straight lines cut one another, not more 
than two (passing) through the same point, and all the points 
(of intersection situated) on one of them be given, and if each of 
those which are on another (of them) lie on a straight line given 
in position — 
or still more generally thus : 

if any number of straight lines cut one another, not more than 

two (passing) through the same point, and all the points (of 

intersection situated) on one of them be given, while of the other 

points of intersection in multitude equal to a triangular number 

a number corresponding to the side of this triangular number lie 

respectively on straight lines given in position, provided that of 

these latter points no three are at the angular points of a triangle 

(«. having for sides three of the given straight lines) — each of the 

remaining points will He on a straight line given in position'. 

■ It is probable that fhe writer of the Elements was not unaware 

of this but that he only set out the principle ; and he seems, in the 

case of all the porisms, to have laid down the principles and the 

seed only [of many important things], the kinds of which should be 

distinguished according to the differences, not of their hypotheses, but 

of the results and the things sought [All the hypotheses are different 

from one another because they are entirely special, but each of the 

results and things sought, being one and the same, follow from many 

different hypotheses.] 

"We must then in the first book distinguish the following kinds of 
things sought : 

"At the beginning of the book' is this proposition : 

I. ' If from two given points straight tines be drawn meeting 
on a straight line given in position, and one cut off from a straight 
line given in position (a segment measured} to a given point on it, 
the other will also cut off from another {straight line a segment) 
liaving to the first a given ratio' 

" Following on this (we have to prove) 

II. that such and such a point lies on a straight line given 

in position ; 
III. that the ratio of such and such a pair of straight lines 
is given ; " 
etc. etc, (up to xxix.). 
"The three books of the porisms contain 38 lemmas; of the 
theorems themselves there are 171." 

1 Loria (p. as6, ». 3) gives the meaning of this as follows, pointing out that Sim son vu 
the discoverer of it : " If a complete n-lateral be deformed so that its sides respectively turn 
about n points on a straight line, and {» ~ 1) of iEs n (« - i)/i vertices move on as many 
straight lines, the other (» - [)(»- j)/i of its vertices likewise move on as many straight 
lines ; but it is necessary that it should be impossible to form with the {ft - 1) vertices any 
triangle having for sides the sides of the polygon." 

1 Reading, with Heiberg, roB pipMov [tuS f Hulischl. 



ch. n] EUCLID'S OTHER WORKS 13 

Pappus further gives lemmas to the Porisms (pp. 866 — 918, ed. 
Hultsch). 

With Pappus' account of Forisms must be compared the passages 
of Proclus on the same subject Proclus distinguishes two senses in 
which the word ir6pierfia is used. The first is that of corollary where 
something appears as an incidental result of a proposition, obtained 
without trouble or special seeking, a sort of bonus which the investi- 
gation has presented us with 1 . The other sense is that of Euclid's 
Porisms*, In this sense* "porism is the name given to things which 
are sought, but need some finding and are neither pure bringing into 
existence nor simple theoretic argument. For (to prove) that the 
angles at the base of isosceles triangles are equal is a matter of 
theoretic argument, and it is with reference to things existing that 
such knowledge is (obtained). But to bisect an angle, to construct a 
triangle, to cut off, or to place — all these things demand the making 
of something ; and to find the centre of a given circle, or to find the 
greatest common measure of two given commensurable magnitudes, 
or the like, is in some sort between theorems and problems. For in 
these cases there is no bringing into existence of the things sought, 
but finding of them, nor is the procedure purely theoretic. For it is 
necessary to bring that which is sought into view and exhibit it to 
the eye. Such are the porisms which Euclid wrote, and arranged in 
three books of Porisms. 

Proclus' definition thus agrees well enough with the first, " older," 
definition of Pappus. A porism occupies a place between a theorem 
and a problem ; it deals with something already existing, as a theorem 
does, but has to find it (e.g. the centre of a circle), and, as a certain 
operation is therefore necessary, it partakes to that extent of the 
nature of a problem, which requires us to construct or produce some- 
thing not previously existing. Thus, besides III. 1 of the Elements 
and X. 3, 4 mentioned by Proclus, the following propositions are 
real porisms: III. 25, VL 11— 13, vn, 33, 34, 36, 39, vm. 2, 4, x. 10, 
XIII. 18. Similarly in Archimedes On the Sphere and Cylinder I. 2 — 6 
might be called porisms. 

The enunciation given by Pappus as comprehending ten of Euclid's 
propositions may not reproduce the form of Euclid's enunciations ; 
but, comparing the result to be proved, that certain points lie on 
straight lines given in position, with the class indicated by II. above, 
where the question is of such and such a point lying on a straight line 
given in position, and with other classes, e.g. (v.) that such and such a 
line is given in position, (VI.) that such and such a line verges to a given 
point, (XXVII.) that there exists a given point such that straight lines 
drawn from it to such and such (circles) will contain a triangle given 
in species, we may conclude that a usual form of a porism was " to 
prove that it is possible to find a point with such and such a property" 

1 Produsjpp. 111, 14; 301, 11. 

■ ibid. p. ill, 11. "The term porism is used of certain problems, like the Ptrrisms 
written by Euclid." 

' ibid. pp. joi, 15 sqq. 



i 4 INTRODUCTION [ch. it 

or "a straight line on which lie all the points satisfying given 
conditions " etc. 

Sim son defined a porism thus : " Porisma est propositi o in qua 
proponitur demonstrate rem aliquant, vel pi u res datas esse, cui, vet 
quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae 
ad ea quae data sunt eandem habent re lat ion em, con venire ostendendum 
est affectionem quandam communem in propositione descriptam 1 ." 

From the above it is easy to understand Pappus' statement that 
loci constitute a large class of porisms. A locus is well defined by 
Simson thus : " Locus est propositio in qua propositum est datam 
esse demonstrare, vel invenire lineam aut superficiem cuius quodlibet 
punctum, vel superficiem in qua quaelibet linea data lege descripta, 
communem quandam ha bet proprietatem in propositione descriptam," 
Heiberg cites an excellent instance of a loats which is a porism, namely 
the following proposition quoted by Eutocius' from the Plane Loci of 
Apollonius : 

" Given two points in a plane, and a ratio between unequal straight 
lines, it b possible to draw, in the plane, a circle such that the straight 
lines drawn from the given points to meet on the circumference of 
the circle have (to one another) a ratio the same as the given ratio," 

A difficult point, however, arises on the passage of Pappus, which 
says that a porism is " that which, in respect of its hypothesis, falls 
short of a locus-theorem " (rovacov StrnptftuiTos:). Heiberg explains it 
by comparing the porism from Apollonius' Plane Loci just given with 
Pappus' enunciation of the same thing, to the effect that, if from two 
given points two straight lines be drawn meeting in a point, and these 
straight lines have to one another a given ratio, the point will lie on 
either a straight line or a circumference of a circle given in position. 
Heiberg observes that in this latter enunciation something is taken 
into the hypothesis which was not in the hypothesis of the enunciation 
of the porism, viz. " that the ratio of the straight lines is the same." 
I confess this does not seem to me satisfactory : for there is no real 
difference between the enunciations, and the supposed difference in 
hypothesis is very like playing with words. Chasles says : " Ce qui 
constitue le porisme est ce qui manque a I'hypotklse d'un tkioreme 
local (en d'autres termes, le porisme est inferieur, par l'hypothese, au 
th^oreme local; e'est-a-dire que quand quelques parties d'une pro- 
position locale n'ont pas dans l'enonce la determination qui ieur est 
propre, cette proposition cesse d'etre regardee comme un th^oreme et 
devient un porisme)." But the subject still seems to require further 
elucidation. 

While there is so much that is obscure, it seems certain (i) that the 
Porisms were distinctly part of higher geometry and not of elementary 

1 This was thus expressed by Chasles : n Le porisme est une proposition dans laquelle on 
demande de demontrer qu'une chose ou plusieurs choses sont denrUts, qui, ainsi que Tone 

3uelconque d'une infinite d'autres choses non donnees, jnais dont chacune est avec des choses 
onntes dans une meme relation, ont une certaine propnete commune, decrite dans 1ft pro- 
position." 

* Commentary on Apollonius" Conies (vol. tt. p. ISO, ed. Heiberg). 



ch. ii] EUCLID'S OTHER WORKS 15 

geometry, (a) that they contained propositions belonging to the 
modern theory of transversals and to projective geometry. It should 
be remembered too that it was in the course of his researches on this 
subject that Chasles was led to the idea of anharmonk ratios. 

Lastly, allusion should be made to the theory of Zeuthen 1 on the 
subject of the porisms. He observes that the only porjsm of which 
Pappus gives the complete enunciation, " If from two given points 
straight lines be drawn meeting on a straight line given in position, 
and one cut off from a straight line given in position (a segment 
measured) towards a given point on it, the other will also cut off from 
another (straight line a segment) bearing to the first a given ratio," 
is also true if there be substituted for the first given straight line a 
conic regarded as the "locus with respect to four lines," and that this 
extended porism can be used for completing Apollonius' exposition 
of that locus. Zeuthen concludes that the Porisms were in part by- 
products of the theory of conies and in part auxiliary means for the 
study of conies, and that Euclid called them by the same name as 
that applied to corollaries because they were corollaries with respect to 
conies. But there appears to be no evidence to confirm this conjecture. 

S- The Surface4oci (tottoi -jrooi hirt^iaveia). 

The two books on this subject are mentioned by Pappus as part 
of the Treasury of Analysis 1 . As the other works in the list which 
were on plane subjects dealt only with straight lines, circles, and 
conic sections, it is a priori likely that among the loci in this treatise 
(loci which are surfaces) were included such loci as were cones, 
cylinders and spheres. Beyond this all is conjecture based on two 
lemmas given by Pappus in connexion with the treatise. 

(1) The first of these lemmas 1 and the figure attached to it are 
not satisfactory as they stand, but a possible restoration is indicated 
by Tannery*. If the latter is right, it suggests that one of the loci 
contained all the points on the elliptical parallel sections of a cylinder 
and was therefore an oblique circular cylinder. Other assumptions 
with regard to the conditions to which the lines in the figure may be 
subject would suggest that other loci dealt with were cones regarded 
as containing all points on particular elliptical parallel sections of 
the cones", 

(2) In the second lemma Pappus states and gives a complete proof 
of the focus-and-directrix property of a conic, viz. that the locus of a 
point whose distance front a given point is in a given ratio to its distance 
from a fixed line is a conic section, which is an ellipse, a parabola or a 
hyperbola according as the given ratio is less than, equal to, or greater 
than unity'. Two conjectures are possible as to the application of 
this theorem in Euclid's Surface-loci, {a) It may have been used to 
prove that the locus of a point whose distance from a given straight 

1 Die Lekrcvwt den Kegilsiknitttn im Aliertttm, chapter VIII. 

* Pappus, vn. p. 636. " Hid- TO p- 1004. 

* Mullciin da seieneei math, eJ astro*., J* Sirie, VI. 1+9. 

1 Further particulars will be found in The Works ef Archimedes, pp. litii — Ixiv, and in 
Zeuthen, Die Lthre von den KtgeUthttitun, p. 415 sqq. 

' Pappus, Vtt. pp. j 006 — 1014, and Hultsdi's Appendix, pp. 1170 — 3. 



1 6 INTRODUCTION [en. 11 

line is in a given ratio to its distance from a given plane is a certain 
cone, (b) It may have been used to prove that the locus of a point 
whose distance from a given point is in a given ratio to its distance 
from a given plane is the surface formed by the revolution of a conic 
about its major or conjugate axis 1 . Thus Chasles may have been 
correct in his conjecture that the Surface-loci dealt with surfaces of 
revolution of the second degree and sections of the same". 

6. The Conks. 

Pappus says of this lost work: "The four books of Euclid's Conies 
were completed by Apollonius, who added four more and gave us 
eight books of Conies'." It is probable that Euclid's work was lost 
even by Pappus' time, for he goes on to speak of "Aristaeus, who wrote 
the still extant five books of Solid Loci connected with the conies." 
Speaking of the relation of Euclid's work to that of Aristaeus on conies 
regarded as loci, Pappus says in a later passage (bracketed however 
by Huttsch) that Euclid, regarding Aristaeus as deserving credit for 
the discoveries he had already made in conies, did not (try to) 
anticipate him or construct anew the same system. We may no 
doubt conclude that the book by Aristaeus on solid loci preceded 
Euclid's on conies and was, at least in point of originality, more 
important Though both treatises dealt with the same subject-matter, 
the object and the point of view were different ; had they been the 
same, Euclid could scarcely have refrained, as Pappus says he did, 
from attempting to improve upon the earlier treatise. No doubt 
Euclid wrote on the general theory of conies as Apollonius did, but 
confined himself to those properties which were necessary for the 
analysis of the Solid Loci of Aristaeus. The Conks of Euclid were 
evidently superseded by the treatise of Apollonius. 

As regards the contents of Euclid's Conks, the most important 
source of our information is Archimedes, who frequently refers to 
propositions in conies as well known and not needing proof, adding 
in three cases that they are proved in the " elements of conies " or in 
"the conies," which expressions must clearly refer to the works of 
Aristaeus and Euclid 4 

Euclid still used the old names for the conies (sections of a right- 
angled, acute 'angled, or obtuse-angled cone), but he was aware that 
an ellipse could be obtained by cutting a cone in any manner by a 
plane not parallel to the base (assuming the section to lie wholly 
between the apex of the cone and its base) and also by cutting a 
cylinder. This is expressly stated in a passage from the Pkaettontena 
of Euclid about to be mentioned". 

7. The Pkaenometta. 

This is an astronomical work and is still extant. A much inter- 

1 For further details see The Works of Arckimtd(s x pp. Ixiv, Lev, and Zeuthen, I. c. 

* Apcrfu hhtoriqtu, pp. 373 — 4. * Pappus, VII. p. 673. 

* For details of these propositions see my Apollonius of Perga, pp. xxxv, xxxvi. 

* Phattamtna, ed. Menge, p. 6: "if a cone or a cylinder be cut by a plane not 
parallel to the base, the section is a section of an acute-angled cone, which is like a shield 
\evftit)." 



ch. n] EUCLID'S OTHER WORKS 17 

polated version appears in Gregory's Euclid. An earlier and better 
recension is however contained in the Ms. Vindobonensis philos. 
Gr. 103, though the end of the treatise, from the middle of prop. 16 
to the last (18), is missing. The book, now edited by Menge 1 , consists 
of propositions in spheric geometry. Euclid based it on Autolycus' 
work Trepi Ktvov/ievTj? ecftatpas, but also, evidently, on an earlier text- 
book of Spkaerica of exclusively mathematical content. It has been 
conjectured that the latter textbook may have been due to Eudoxus'. 

8. The Optics. 

This book needs no description, as it has been edited by Heiberg 
recently', both in its genuine form and in the recension by Theon. 
The Catoptrica published by Heiberg in the same volume is not 
genuine, and Heiberg suspects that in its present form it may be 
Theon's. It is not even certain that Euclid wrote Catoptrica. at all, as 
Proclus may easily have had Theon's work before him and inadvertently 
assigned it to Euclid 1 . 

9. Besides the above-mentioned works, Euclid is said to have 
written the Elements of Music* (at tcara ftovatKrjv a"rotYet<a<r£ts). Two 
treatises are attributed to Euclid in our MSS. of the Mtfsici, the 
icararofii) Kavovos, Sectio canonis (the theory of the intervals), and the 
tta-aiyayTj appovi/ci} (introduction to harmony)*. The first, resting on 
the Pythagorean theory of music, is mathematical, and the style and 
diction as well as the form of the propositions mostly agree with what 
we find in the Elements. Jan thought it genuine, especially as almost 
the whole of the treatise (except the preface) is quoted in extenso, and 
Euclid is twice mentioned by name, in the commentary on Ptolemy's 
Harmonica published by Wallis and attributed by him to Porphyry. 
Tannery was of the opposite opinion'. The latest editor, Menge, sug- 
gests that it may be a redaction by a less competent hand from the 
genuine Euclidean Elements of Music. The second treatise is not 
Euclid's, but was written by Cleonides, a pupil of Aristoxenus*. 

Lastly, it is worth while to give the Arabians' list of Euclid's 
works. I take this from Suter's translation of the list of philosophers 
and mathematicians in the Fikrist, the oldest authority of the kind 
that we possess*. " To the writings of Euclid belong further [in 
addition to the Elements] : the book of Phaenomena ; the book of 



1 Euelidis Optra omnia, vol. VIII., igib, pp. 2 — 156. 

1 Heiberg, Euklid-Studien, p. 46 ; HtilEsch, Autolycut, p. xii ; A. A. Bjornbo, Studien 
iiber Menetaos' Spharik {Abhattdiungen nir Geschichte der wathematiscken tVissenscAaflen, 
Jtiv. 1002), p. ;6sqq. 

3 Euelidis opera omnia, vol. VI [. (1895). 

* Heiberg, Euclid's Optics, tic. p. I. a Proclus, p. 69, 3. 

* Both trealises edited by Jan in Music i Scriptorts Gratci, 1895, pp. 113—166, 167 — 10;, 
and by Menge in Euelidis opera omnia, vol. VII]., 1916, pp. 157 — 183, 1S5 — 223. 

7 Cpmptes rendus de VAcad. des inscriptions it belles-lettres, Paris, 1904, pp. +39 — 445. 
Cf. Bihtioiheea Mathtmaiica, VI3, [905-6, p. 225, note i. 

a Heiberg, Euktid-Studien, pp. 52 — 55 \ Jan, Musici Seriptores Graeci, pp. 169—174. 

6 H. Suter, Das Mathematiker- Vcruichniss im Fikrist in Abhandlungen iter Gesckicktc 
der Mathematik, VI., 1892, pp. 1- — 87 (see especially p. 17). Cf. Casiri, I. 339, 340, and 
Gartz, Dc interpretiius et txplanatoribus Euelidis Arabieis, 1823, pp. 4, $• 



i8 INTRODUCTION [ch. n 

Given Magnitudes [Data] ; the book of Tones, known under the name 
of Music, not genuine ; the book of Division, emended by Thabit ; 
the book of Utilisations or Applications [Porisms], not genuine; the 
book of the Canon ; the book of the Heavy and Light ; the book of 
Synthesis, not genuine ; and the book of Analysis, not genuine." 

It is to be observed that the Arabs already regarded the book of 
Tones (by which must be meant the citrayayf} apuovticrj) as spurious. 
The book of Division is evidently the book on Divisions (of figures). 
The next book is described by Casiri as " liber de utilitate suppositus." 
Suter gives reason for believing the Porisms to be meant 1 , but does 
not apparently offer any explanation of why the work is supposed to 
be spurious. The book of the Canon is clearly the KaraTOfit} icavovos. 
The book on "the Heavy and Light" is apparently the tract Dt Uvi 
et ponderoso, included in the Basel Latin translation of 1537, and in 
Gregory's edition. The fragment, however, cannot safely be attributed 
to Euclid, for ( 1 ) we have nowhere any mention of his having written 
on mechanics, (2) it contains the notion of specific gravity in a form 
so clear that it could hardly be attributed to anyone earlier than 
Archimedes*. Suter thinks* that the works on Analysis and Synthesis 
(said to be spurious in the extract) may be further developments of 
the Data or Porisms, or may be the interpolated proofs of Euci. 
XIII. 1 — 5, divided into analysis and synthesis, as to which see the notes 
on those propositions. 

1 Suter, op. tit. pp. 49, 50, Wen rich translated the word as " ulilia. " Suter says thai 
the nearest meaning of the Arabic word as of "porism" is tut, gain (Nutzen, Gewinn), while 
a further meaning is explanation, observation, addition : a gain arising out of what has 
preceded (cf. Prod us 1 definition of the po risen in the sense of a corollary). 

1 Heiberg, Euklid-Stvditn, pp. 0, 10. * Suter, ?p* til. p. 50. 



CHAPTER III. 

GREEK COMMENTATORS ON THE ELEMENTS OTHER 
THAN PROCLUS. 

That there was no lack of commentaries on the Elements before 
the time of Proctus is evident from the terms in which Proclus refers 
to them ; and he leaves as in equally little doubt as to the value 
which, in his opinion, the generality of them possessed. Thus he says 
in one place (at the end of his second prologue) 1 : 

" Before making a beginning with the investigation of details, 
I warn those who may read me not to expect from me the things 
which have been dinned into our ears ad nauseam (St,aT€9pv\*)T<u) by 
those who have preceded me, viz. lemmas, cases, and so forth. For 
I am surfeited with these things and shall give little attention to them. 
But I shall direct my remarks principally to the points which require 
deeper study and contribute to the sum of philosophy, therein emulating 
the Pythagoreans who even had this common phrase for what I mean 
' a figure and a platform, but not a figure and sixpence 1 .' " 

In another place' he says : " Let us now turn to the elucidation 
of the things proved by the writer of the Elements, selecting the more 
subtle of the comments made on them by the ancient writers, while 
cutting down their interminable diffuseness, giving the things which 
are more systematic and follow scientific methods, attaching more 
importance to the working-out of the real subject-matter tharr to the 
variety of cases and lemmas to which we see recent writers devoting 
themselves for the most part." 

At the end of his commentary on Eucl, I. Prockis remarks' that 
the commentaries then in vogue were full of all sorts of confusion, and 
contained no account of causes, no dialectical discrimination, and no 
philosophic thought. 

These passages and two others in which Proclus refers to "the 
commentators*" suggest that these commentators were numerous. 
He does not however give many names; and no doubt the only 
important commentaries were those of Heron, Porphyry, and Pappus. 

1 Proclus, p. 84, 8. 

■ i.e. we reach a certain height, use the platform so attained as a base on which to build 
another stage, then use that as a base and so on. 

' Proclus, p. 300, to. 4 Hid. p. 431, 15. * ibid. p. 389, \\ \ p* 318, r6. 



io INTRODUCTION [ch. hi 

I. Heron. 

Froctus alludes to Heron twice as Heron mecAanicus 1 , in another 
place* he associates him with Ctesibius, and in the three other 
passages ' where Heron is mentioned there is no reason to doubt that 
the same person is meant, namely Heron of Alexandria. The date of 
Heron is still a vexed question. In the early stages of the controversy 
much was made of the supposed relation of Heron to Ctesibius. The 
Dest MS. of Heron's Belopoeica has the heading "Hpowo? Kryvi&iov 
0e\oTrotiicd, and an anonymous Byzantine writer of the tenth century, 
evidently basing himself on this title, speaks of Ctesibius as Heron's 
xafljfyjjTTJf, " master " or " teacher." We know of two men of the name 
of Ctesibius. One was a barber who lived in the time of Ptolemy 
Euergetes II, i.e. Ptolemy VII, called Physcon {died 117 B.C.), and 
who is said to have made an improved water-organ*. The other was a 
mechanician mentioned by Athenaeus as having made an elegant 
drinking-horn in the time of Ptolemy Philadelphus (285-247 B.C.)*. 
Martin* took the Ctesibius in question to be the former and accord- 
ingly placed Heron at the beginning of the first century B.C., say 
1 26-50 B.C But Philo of Byzantium 7 , who repeatedly mentions Ctesi- 
bius by name, says that the first mechanicians had the advantage of 
being under kings who loved fame and supported the arts. Hence our 
Ctesibius is more likely to have been the earlier Ctesibius who was 
contemporary with Ptolemy 1 1 Philadelphus. 

But, whatever be the date of Ctesibius, we cannot safely conclude 
that Heron was his immediate pifpil. The title " Heron's (edition of) 
Ctesibius's Belopoeica" does not, in fact, justify any inferenee as to 
the interval of time between the two works. 

We now have better evidence for a terminus post quern. The 
Metrica of Heron, besides quoting Archimedes and Apollonius, twice 
refers to " the books about straight lines (chords) in a circle " {iv rots 
wept r£* 4» xvtcXqt eu&ei&p). Now we know of no work giving a Table 
of Chords earlier than tbat of Hipparchus. We get, therefore, at 
once, 1 50 B.c. or thereabouts as the terminus post quern. But, again, 
Heron's Meckanica quotes a definition of " centre of gravity " as given 
by " Posidonius, a Stoic " : and, even if this Posidonius lived before 
Archimedes, as the context seems to imply, it is certain that another 
work of Heron's, the Definitions, owes something to Posidonius of 
Apamea or Rhodes, Cicero's teacher (135—51 B.C.). This brings Heron's 
date down to the end of the first century B.C., at least 

We have next to consider the relation, if any, between Heron and 
Vitruvius. In his De Architecture brought out apparently in 14 B.C., 
Vitruvius quotes twelve authorities on machinationes including Archytas 

1 Proclus, p. 305, 14 ; p. 346, 13. 
1 ibid. p. 41, to. 

* ibid. p. 196, 16 ; p. 313, 7 : p- 4*9, 13. 

* Athetuetu, Ddpno-Sefh. iv., c. 7;, p. 174 *— <■ • 

* ibid, xi., c. 97, p. 497 b — e. 

* Martin, KschtriJui mr lavitetUt mtvritgta d'Hiren d ' Altxandrit, Puii, l8j+, p. 17. 
7 Philo, Mi/than. Synt., p. 50, 38, ed. Schone. 



ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS ai 

(second), Archimedes (third), Ctesibius (fourth) and Philo of Byzan- 
tium (sixth), but does not mention Heron. Nor is it possible to 
establish inter-dependence between Vitmvius and Heron ; the differ- 
ences between them seem on the whole more numerous and important 
than the resemblances (e.g. Vitruvius uses 3 as the value of tt, while 
Heron always uses the Archimedean value 3^). The inference is that 
Heron can hardly have written earlier than the first century A.D. 

The most recent theory of Heron's date makes him later than 
Claudius Ptolemy the astronomer (i 00- 1 78 A.D.). The arguments are 
mainly these. (!) Ptolemy claims as a discovery of his own a method 
of measuring the distance between two places (as an arc of a great 
circle on the earth's surface) in the case where the places are neither 
on the same meridian nor on the same parallel circle. Heron, in his 
Dtoptra, speaks of this method as of a thing generally known to 
experts. (2) The dioptra described in Heron's work is a fine and 
accurate instrument, much better than anything Ptolemy had at his 
disposal. (3) Ptolemy, in his work Ilepi poir&v, asserted that water 
with water round it has no weight and that the diver, however deep 
he dives, does not feel the weight of the water above him. Heron, 
strangely enough, accepts as true what Ptolemy says of the diver, but 
is dissatisfied with the explanation given by "some," namely that it is 
because water is uniformly heavy — this seems to be equivalent to 
Ptolemy's dictum that water in water has no weight — and he essays a 
different explanation based on Archimedes. (4) It is suggested that 
the Dionysius to whom Heron dedicated his Definitions is a certain 
Dionysius who was firaefectus itrbi in 301 A.D. 

On the other hand Heron was earlier than Pappus, who was 
writing under Diocletian (284-305 A.D.), for Pappus alludes to and 
draws upon the works of Heron. The net result, then, of the most 
recent research is to place Heron in the third century a.d. and perhaps 
little earlier than Pappus. Heiberg 1 accepts this conclusion, which 
may therefore, perhaps, be said to hold the field for the present*. 

That Heron wrote a systematic commentary on the Elements 
might be inferred from Proclus, but it is rendered quite certain by 
references to the commentary in Arabian writers, and particularly in 
an-Nairlzi's commentary on the first ten Books of the Elements. The 
Fihrist says, under Euclid, that " Heron wrote a commentary on this 
book [the Elements], endeavouring to solve its. difficulties' " ; and 
under Heron, " He wrote : the book of explanation of the obscurities 
in Euclid'...." An-Nairfzl's commentary quotes Heron by name very 
frequently, and often in such a way as to leave no doubt that the 
author had Heron's work actually before him. Thus the extracts are 
given in the first person, introduced by "Heron says" ("Dixit Yrinus" 



1 Htronis Alexandrini o/vra t vol. V. (Teuhner, 1*914), p. ix. 

- Fuller details of the various arguments will be found in my History of Greek Maiht- 
matin, toil, vol. it., pp. j 96 —306. 

3 Das Matkematiker.VeneitAniss im Fihrist (tr. Sutcr), p. 16. 
* Hid. p. 11, 



t a INTRODUCTION [ch. in 

or " Heron ") ; and in other places we are told that Heron " says 
nothing," or * is not found to have said anything," on such and such 
a proposition. The commentary of an-Nairlzi is in part edited by 
Besthorn and Heiberg from a Leiden MS. of the translation of the 
Elements by al-Hajjaj with the commentary attached 1 . But this MS. 
only contains six Books, and several pages in the first Book, which 
contain the comments of Simplicius on the first twenty-two defini- 
tions of the first Book, are missing. Fortunately the commentary of 
an-Nairlzi has been discovered in a more complete form, in a Latin 
translation by Gherardus Cremonensis of the twelfth century, which 
contains the missing comments by Simplicius and an-Nairlzl's com- 
ments on the first ten Books. This valuable work has recently been 
edited by Curtze'. 

Thus from the three sources, P rod us, and the two versions of 
an-Nairlzi, which supplement one another, we are able to form a very 
good idea of the character of Heron's commentary. In some cases 
observations given by Proclus without the name of their author are 
seen from an-Nairizi to be Heron's ; in a few cases notes attributed 
by Procius to Heron are found in an-Nairizi without Heron's name; 
and, curiously enough, one alternative proof (of I. 25) given as Heron's 
by Proclus is introduced by the Arab with the remark that he has 
not been able to discover who is the author. 

Speaking generally, the comments of Heron do not seem to have 
contained much that can be called important. We find 

(1) A few general notes, e.g. that Heron would not admit more 
than three axioms. 

(2) Distinctions of a number of particular cases of Euclid's pro- 
positions according as the figure is drawn in one way or in another. 

Of this class are the different cases of 1. 35, 36, HI. 7, 8 (where the 
chords to be compared are drawn on different sides of the diameter 
instead of on the same side), III. 12 (which is not Euclid's, but Heron's 
own, adding the case of external contact to that of internal contact in 
III. 1 1), VI. 19 (where the triangle in p which an additional line is drawn 
is taken to be the smaller of the two), Vii. 19 (where he gives the 
particular case of three numbers in continued proportion, instead of 
four proportionals). 

(3) Alternative proofs. Of these there should be mentioned (a) 
the proofs of H. I — 10 "without a figure," being simply the algebraic 
forms of proof, easy but un instructive, which are so popular nowadays, 
the proof of III. 25 (placed after III. 30 and starting from the arc 
instead of the chord), III. 10 (proved by 111. 9), 111. 13 (a proof 
preceded by a lemma to the effect that a straight line cannot meet a 
circle in more than two points). Another class of alternative proof is 



1 Cedes Leidtnris 399, r. Eudidis Eiementa ex interprets itme ai-ffadschdiehadsekii 
cum commentariis ai-Naritii. Five parts carrying I he work Lo the end of Book IV. were 
issued in 1893. 1897, J900, 1905 awl 1910 respectively. 

* Anaritii in decern Hbros prieres elementorum Eudidis eonimtntarii ex ittterpreiaiione 
Gherardi Cremonensis... edidit Maximilian us Curtze (Teubner, Leipzig, 1899}. 



ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS 33 

(6) that which is intended to meet a particular objection («Wra«-«) 
which had been or might be raised to Euclid's construction. Thus 
in certain cases he avoids producing a particular straight line, where 
Euclid produces it, in order to meet the objection of any one who should 
deny our right to assume that there is any space available^. Of this 
class are Heron's proofs of I. 1 1, 1, 20, and his note on I. 16. Similarly 
on I. 48 he supposes the right-angled triangle which is constructed to 
be constructed on the same side of the common side as the given 
triangle is. A third class (c) is that which avoids redttctio ad 
absurdum. Thus, instead of indirect proofs, Heron gives direct 
proofs of I. 19 (for which he requires, and gives, a preliminary 
lemma), and of I. 25. 

(4) Heron supplies certain converses of Euclid's propositions, 
e.g. converses of II. 12, 13, vm. 27. 

(5) A few additions to, and extensions of, Euclid's propositions 
are also found. Some are unimportant, e.g. the construction of isosce'es 
and scalene triangles in a note on I. 1 , the construction of tivo tangents 
in in. 17, the remark that vn. 3 about finding the greatest common 
measure of three numbers can be applied to as many numbers as we 
please (as Euclid tacitly assumes in VII. 31). The most important 
extension is that of in. 20 to the case where the angle at the 
circumference is greater than a right angle, and the direct deduction 
from this extension of the result of III. 22. Interesting also are the 
notes on I. 37 (on I. 24 in Proclus), where Heron proves that two 
triangles with two sides of one equal to two sides of the other and 
with the included angles supplementary are equal, and compares the 
areas where the sum of the two included angles (one being supposed 
greater than the other) is less or greater than two right angles, and 
on I, 47, where there is a proof (depending on preliminary lemmas) of 
the fact that, in the figure of the proposition, the straight lines AL, 
BK, CF meet in a point. After IV. 16 there is a proof that, in a 
regular polygon with an even number of sides, the bisector of one 
angle also bisects its opposite, and an enunciation of the corresponding 
proposition for a regular polygon with an odd number of sides. 

Van Peach" gives reason for attributing to Heron certain other 
notes found in Proclus, viz. that they are designed to meet the same 
sort of points as Heron had in view in other notes undoubtedly written 
by him. These are (a) alternative proofs of I. 5, I. 17, and I. 32, 
which avoid the producing of certain straight lines, (b) an alternative 
proof of I. 9 avoiding the construction of the equilateral triangle on 
the side of BC opposite to A ; {c) partial converses of 1. 35 — 38, starting 
from the equality of the areas and the fact of the parallelograms or 
triangles being in the same parallels, and proving that the bases are 
the same or equal, may also be Heron's. Van Pesch further supposes 
that it was in Heron's commentary that the proof by Menelaus of 
I, 25 and the proof by Philo of I. 8 were given. 

1 Cf. Proclui, it;., 1 tlU W701 tii rlrxov pj) eliivni..., 189, 18 \tyct oir Tti Sri ott tm 
riwn,,., 

3 Dt Prpcli jontibusi Lugduni-fiatavomm, 1900, 



a 4 INTRODUCTION [ch. hi 

The last reference to Heron made by an-NairtzI occurs in the note 
on VIII. 27, so that the commentary of the former must at least have 
reached that point. 

II. Porphyry. 

The Porphyry here mentioned is of course the Neo-Flatonist who 
lived about 232-304 a.d. Whether he really wrote a systematic 
commentary on the Elements is uncertain. The passages in Proclus 
which seem to make this probable are two in which he mentions him 
( 1 ) as having demonstrated the necessity of the words " not on the 
same side " in the enunciation of I. 14 1 , and {2) as having pointed out 
the necessity of understanding correctly the enunciation of I, 26, since, 
if the particular injunctions as to the sides of the triangles to be taken 
as equal are not regarded, the student may easily fall into error*. 
These passages, showing that Porphyry carefully analysed Euclid's 
enunciations in these cases, certainly suggest that his remarks were 
part of a systematic commentary. Further, the list of mathematicians 
in the Fihrist gives Porphyry as having written "a book on the 
Elements." It is true that Wen rich takes this book to have been a 
work by Porphyry mentioned by Suidas and Proclus ( Theolog. Platan.), 
•jrtpl dpx&v libri II.' 

There is nothing of importance in the notes attributed to Porphyry 
by Proclus. 

( 1) Three alternative proofs of I. 20, which avoid producing a side 
of the triangle, are assigned to Heron and Porphyry without saying 
which belonged to which. If the first of the three was Heron's, I 
agree with van Pesch that it is more probable that the two others 
were both Porphyry's than that the second was Heron's and only the 
third Porphyry's. For they are similar in character, and the third 
uses a result obtained in the second*. 

(2) Porphyry gave an alternative proof of I. 18 to meet a childish 
objection which is supposed to require the part of A C equal to A B to 
be cut off from CA and not from A C. 

Proclus gives a precisely similar alternative proof of I. 6 to meet a 
similar supposed objection ; and it may well be that, though Proclus 
mentions no name, this proof was also Porphyry's, as van Pesch 
suggests*. 

Two other references to Porphyry found in Proclus cannot have 
anything to do with commentaries on the Elements. In the first a 
work called the Xv^fuxrd is quoted, while in the second a philo- 
sophical question is raised. 

III. Pappus. 

The references to Pappus in Proclus are not numerous; but we 
have other evidence that he wrote a commentary on the Elements. 
Thus a scholiast on the definitions of the Data uses the phrase " as 

1 Proclus, pp. 197, 1—198, 10. ' ibid, p. 351, 13, 141ml the pages preceding. 

* Fihrist (tr. Suter), p. 9, jo and p. 45 {note j). 

* Van Pesch, Di Pretli fcntibus, pp. no, 130. Heibetg assigned them as above in hit 
Eu&lid-Studitn (p. 160), but seems to have changed hit view later. (See Besthom- Heiberg, 
Codex Ltidatsis, p. 93, note ».) 

* Van Pesch, sp. cii. pp. 130 — \. 



ch. tit] GREEK COMMENTATORS OTHER THAN PROCLUS as 

Pappus says at the beginning of his (commentary) on the loth (book) 
of Euclid 1 ." Again in the Fihrist we are told that Pappus wrote a 
commentary to the tenth book of Euclid in two parts'. Fragments 
of this still survive in a MS. described by Woepcke*, Paris. No. 952. 2 
(supplement arabe de la Bibliotheque imperiale), which contains a 
translation by Abu 'Uthman (beginning of [Oth century) of a Greek 
commentary on Book X. It is in two books, and there can now be 
no doubt that the author of the Greek commentary was Pappus 4 . 
Again Eutocius, in his note on Archimedes, On the Sphere and 
Cylinder I. r3, says that Pappus explained in his commentary on the 
Elements how to inscribe in a circle a polygon similar to a polygon 
inscribed in another circle ; and this would presumably come in his 
commentary on Book XII., just as the problem is solved in the second 
scholium on Eucl, XII. 1. Thus Pappus' commentary on the Elements 
must have been pretty complete, an additional confirmation of this 
supposition being forthcoming in the reference of Marin us (a pupil 
and follower of Proclus) in his preface to the Data to " the com- 
mentaries of Pappus on the book*." 

The actual references to Pappus in Proclus are as follows : 

(1) On the Postulate (4) that all right angles are equal, Pappus is 
quoted as saying that the converse, viz. that all angles equal to a 
right angle are right, is not true', since the angle included between 
the arcs of two semicircles which are equal, and have their diameters 
at right angles and terminating at one point, is equal to a right angle, 
but is not a right angle. 

(2) On the axioms Pappus is quoted as saying that, in addition to 
Euclid's axioms, others are on record as well (ovvavarfpafao-Sai) about 
unequals added to equals and equals added to unequals 1 ; these, says 
Proclus, follow from the Euclidean axioms, while others given by 
Pappus are involved by the definitions, namely those which assert 
that "all parts of the plane and of the straight line coincide with one 
another," that " a point divides a straight line, a line a surface, and a 
surface a solid," and that "the infinite is (obtained) in magnitudes 
both by addition and diminution*." 

1 Euclid's Data, ed. Menge. p. 761. ' fihrist (tr. Sitter), p. It, 

* JlfJmeires prlsenth A Faeadimit ties jcieneej, 1856, Kiv. pp. 658 — 7(0. 

* Woepcke read the name of the author, in the Litle of the first book, as B.tes (the dot 
representing 11 missing vowel). He quotes also from other MSS. (e.g. of the Ta*rikh al- 
HttlamA and of the Fihrist) when he reads the name of the commentator as B .lis, B .n.t 
at B.I.j, Woepcke takes this author to be Valens, and thinks it possible that he may be 
the same as the astrologer Vettius Valens. This Heiberg (Eu&tiii-Studun, pp. 169, 170) 
proves to be impossible, because, while one of the mss. quoted by Woepcke says that 
"B.n.j, le Rs&mV (late- Greek) was later than Claudius Ptolemy and the Fihrist says 
11 B.I. j, le Betsmt" wrote a commentary on Ptolemy's Plastisphaerium, Vettius Valens 
seems to have Lived under Hadrian, and must therefore have been an elder contemporary of 
Ptolemy. But Suter shows (Fihrist, p. at and p. j*, note 91) that Bants is only distin- 
guished from Babe j by the position of a certain dot, and Bates may also easily have arisen 
from an original Boies (there is no P in Arabic), so that Pappus must be the person meant. 
This is further confirmed by the fact that the Fihrist gives this author and Valens as the 
subjects of twu separate paragraphs, attributing to the latter astrological works only. 

* Heiberg, Euklid-Studien, p. 173; Euclid's Data, ed. Menge, pp. 356, lii. 

* Proclus, pp. 189, jqo. 7 ibid. p. 197, 6—10. 

* Hid. p. 198, 3— ij. 



16 INTRODUCTION [ch, hi 

(3) Pappus gave a pretty proof of I. 5. This proof has, I think, 
been wrongly understood ; on this point see my note on the 
proposition. 

(4) On I. 47 Proclus says 1 : " As the proof of the writer of the 
Elements is manifest, I think that it is not necessary to add anything 
further, but that what has been said is sufficient, since indeed those 
who have added more, like Heron and Pappus, were obliged to make 
use of what is proved in the sixth book, without attaining any 
important result." We shall see what Heron's addition consisted of; 
what Pappus may have added we do not know, unless it was some- 
thing on the lines of his extension of I. 47 found in the Synagoge 
(iv. p. 176, ed. Hultsch). 

We may fairly conclude, with van Pesch", that Pappus is drawn 
upon in various other passages of Proclus where he quotes no 
authority, but where the subject-matter reminds us of other notes 
expressly assigned to Pappus or of what we otherwise know to have 
been favourite questions with him. Thus : 

[. We are reminded of the curvilineal angle which is equal to but 
not a right angle by the note on I. 32 to the effect that the converse 
(that a figure with its interior angles together equal to two right 
angles is a triangle) is not true unless we confine ourselves to 
rectilineal figures. This statement is supported by reference to a 
figure formed by four semicircles whose diameters form a square, and 
one of which is turned inwards while the others are turned outwards. 
The figure forms two angles "equal to" right angles in the sense 
described by Pappus on Post. 4, while the other curvilineal angles are 
not considered to be angles at all, and are left out in summing the 
internal angles. Similarly the allusions in the notes on I. 4, 23 to 
curvilineal angles of which certain moon -shaped angles {i^vouZwi) 
are shown to be "equal to" rectilineal angles savour of Pappus. 

2. On 1. 9 Proclus says* that " Others, starting from the Archi- 
medean spirals, divided any given rectilineal angle in any given ratio." 
We cannot but compare this with Pappus IV. p. 286, where the spiral 
is so used ; hence this note, including remarks immediately preceding 
about the conchoid and the quad rat rix, which were used for the same 
purpose, may very well be due to Pappus. 

3, The subject of isoperi metric figures was a favourite one with 
Pappus, who wrote a recension of Zenodorus' treatise on the subject 1 . 
Now on I. 35 Proclus speaks 1 about the paradox of parallelograms 
having equal area (between the same parallels) though the two sides 
between the parallels may be of any length, adding that of parallelo- 
grams with equal perimeter the rectangle is greatest if the base be 
given, and the square greatest if the base be not given etc. He 
returns to the subject on I. 37 about triangles'. Compare 7 also his 
note on I. 4. These notes may have been taken from Pappus. 

1 Proclus, p. 439, — 15. 

9 Vin Pesch, Dt Prodi fsntitws, p. 13+ sqq. ' Proclus, p. 371, 10. 

* Pappus, v. pp. 304 — 330 ; for Zenodorus own treatise see Hultsch's Appendix, pp. 1 189 
— 1*11. 

* Proclus, pp. 396—8. ■ ibvl, pp. 403— 4. T ibid. pp. 136—7. 



ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS 27 

4. Again, on I. 21, Procius remarks on the paradox that straight 
lines may be drawn from the base to a point within a triangle which 
are (1) together greater than the two sides, and (2) include a less 
angle, provided that the straight lines may be drawn from points in 
the base other than its extremities. The subject of straight lines 
satisfying condition (1) was treated at length, with reference to a 
variety of cases, by Pappus 1 , after a collection of "paradoxes" by 
Erycinus, of whom nothing more is known. Procius gives Pappus' 
first case, and adds a rather useless proof of the possibility of drawing 
straight lines satisfying condition {2) alone, adding that " the proposi- 
tion stated has been proved by me without using the parallels of 
the commentators'." By "the commentators" Pappus is doubtless 
meant 

5. Lastly, the "four-sided triangle," called by Zenodorus the 
" hollow-angled,"* is mentioned in the notes on I. Def. 24 — 29 and 
t. 21. As Pappus wrote on Zenodorus' work in which the term 
occurred*, Pappus may be responsible for these notes. 

IV. Simplicius. 

According to the FihrisP, Simplicius the Greek wrote "a com- 
mentary to the beginning of Euclid's book, which forms an introduc- 
tion to geometry." And in fact this commentary on the definitions, 
postulates and axioms (including the postulate known as the Parallel- 
Axiom) is preserved in the Arabic commentary of an-Nairizi*. On 
two subjects this commentary of Simplicius quotes a certain "Aganis," 
the first subject being the definition of an angle, and the second the 
definition of parallels and the parallel-postulate. Simplicius gives 
word for word, in a long passage placed by an-Nairizi after I. 29, an 
attempt by " Aganis " to prove the para lie I -postulate. It starts from 
a definition of parallels which agrees with Geminus' view of them as 
given by Procius', and is closely connected with the definition given 
by Posidonius 8 . Hence it has been assumed that " Aganis " is none 
other than Geminus, and the historical importance of the commentary 
of Simplicius has been judged accordingly. But it has been recently 
shown by Tannery that the identification of "Aganis" with Geminus 
is practically impossible 11 . In the translation of Besthorn-Heiberg 
Aganis is called by Simplicius in one place "philosophus Aganis," in 
another " magister noster Aganis," in Gherard's version he is " socius 
Aganis" and "socius noster Aganis." These expressions seem to 
leave no doubt that Aganis was a contemporary and friend, if not 
master, of Simplicius ; and it is impossible to suppose that Simplicius 
(fl. about 500 a.d.) could have used them of a man who lived four and 

1 Pappus, HI. pp. 104 — 1 jo. 3 Procius, p. 318, 15. 

1 Procius, p. |6{, »+; ef. pp. 318, 319. * See Pappus, ed. Hultsch, pp. 1154, iip6. 

1 Fihrist (tr. Suter), p. 11. 

1 An-NairM, ed. Bestbom-Heiberg, pp. 9 — 41, 119— 133, ed. Curtie, pp. 1 — 37,65 — 73. 
The Cedtx Ltittensis, from which Besthorn and Heiberg's edition is taken, has unfortunately 
lost some leaves, so that there is a gap from Def. 1 to Def. 13 (parallels). The loss is, how- 
ever, made good by Curtse's edition of the translation by Gherard of Cremona. 

7 Procius, p. 177, 11. 8 ibid, p. 176, 7. 

* Bibliothtca Afathtmatka, ITj, 1900, pp. o — 11. 



28 INTRODUCTION [ch. hi 

a half centuries before his time. A phrase in Simplicius' word-for- 
word quotation from Aganis leads to the same conclusion. He speaks 
of people who objected " even in ancient times " (iam antiquitus) to 
the use by geometers of this postulate. This would not have been an 
appropriate phrase had Gem in us been the writer. I do not think 
that this difficulty can he got over by Sulcus suggestion 1 that the 
passages in question may have been taken out of Heron's commentary, 
and that an-Nairlzl may have forgotten to name the author ; it seems 
clear that Simplicius is the person who described " Aganis." Hence 
we are driven to suppose that Aganis was not Geminus, but some 
unknown contemporary of Simplicius 5 . Considerable interest will 
however continue to attach to the comments of Simplicius so 
fortunately preserved. 

Proclus tells us that one Aegaeas {? Aenaeas) of Hi era polls wrote an 
epitome of the Elements* ; but we know nothing more of him or of it 

1 Ztitsckrififiir Math. u. Pkysik, XLIV., hiat.-litt, Ablh. p. fit. 

* The above argument seems to me quite insuperable. The other arguments of Tannery 
do not, however, cany conviction to my mind. I do not follow the reasoning based on 
Aganis' definition of an angle. It appears to me a pure assumption that Geminus would have 
seen that Posidonius' definition of parallels was not admissible. Nor does it seem to me to 
count for much that Proclus, while telling us that Geminus held that the postulate ought to be 
proved and warned the unwary against hastily concluding that two straight lines approaching 
one another must necessarily meet (cf, a curve and its asymptote), gives no hint that 
Geminus did try to prove the postulate. It may well be that Proclus omitted Geminus' 
" proof" (if he wrote one) because he preferred Ptolemy's attempt which he give* 
(pp. 36S — 7r- 

* Proelui, p. 361, *i. 


















CHAPTER IV. 

PROCLUS AND HIS SOURCES'. 

It is well known that the commentary of P roc 1 us on Eucl. Book i. 
is one of the two main sources of information as to the history of 
Greek geometry which we possess, the other being the Collection of 
Pappus. They are the more precious because the original works of 
the forerunners of Euclid, Archimedes and Apollonius are lost, having 
probably been discarded and forgotten almost immediately after the 
appearance of the masterpieces of that great trio. 

Prod us himself lived 410-485 AD, so that there had already 
passed a sufficient amount of time for the tradition relating to the 
pre -Euclidean geometers to become obscure and defective. In this 
connexion a passage is quoted from Simplkius' who, in his account 
of the quadrature of certain lunes by Hippocrates of Chios, while 
mentioning two authorities for his statements, Alexander Aphro- 
disiensis (about 220 A.D.) and Eudemus, says in one place *, " As 
regards Hippocrates of Chios we must pay more attention to Eudemus, 
since he was nearer the times, being a pupil of Aristotle:" 

The importance therefore of a critical examination of Proclus' 
commentary with a view to determining from what original sources 
he drew need not be further emphasised. 

Proclus received his early training in Alexandria, where Olympio- 
dorus was his instructor in the works of Aristotle, and mathematics 
was taught him by one Heron' (of course a different Heron from the 
•' mechanicus Hero" of whom we have already spoken). He after- 
wards went to Athens where he was imbued by Plutarch, and by 
Syrianus, with the Neo- Platonic philosophy, to which he then devoted 

1 My task in this chapter is nude easy by the appearance, in the nick of time, of the 
dissertation Dt Froili fontibus by J. G, van Pescb (Lugduni-Batavoruro, A pud L. van 
Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough 
acquaintance on the part of the author with alt the literature hearing on it; he coven 
the whole field and he exercises a sound and sober judgment in forming his conclusion s. 
The same cannot always be said of his only predecessor in the same inquiry, Tannery 
(in La Ghmitrit grtcqut, 1887), who often robs his speculations of much of their value 
through his proneness to run away with on idea; be docs so in ihi* case, basing most of his 
conclusions on an arbitrary and unwarranted assumption as to the significance of the words 
ot wtpt rtira (e.g. 'Hpvra, UoettSwrior etc) as used in Proclus. 

■ Simplicius on Aristotle's Phyiits, ed. Diels, pp. j+ — 69. 

' Hid. p. 68, 31. 

4 Cf. Martin, Rtsktrchts atr la vu if la nonages d'Hinm iTAltxandrit, pp. 140 — 1. 

* 



3 o INTRODUCTION [ch. iv 

heart and soul, becoming one of its most prominent exponents. He 
speaks everywhere with the highest respect of his masters, and 
was in turn regarded with extravagant veneration by his contem- 
poraries, as we learn from Marinus his pupil and biographer. On 
the death of Syrianus he was put at the head of the Neo-Platonic 
school. He was a man of untiring industry, as is shown by the 
number of books' which he wrote, including a large number of com- 
mentaries, mostly on the dialogues of Plato. He was an acute 
dialectician, and pre-eminent among his contemporaries in the 
range of his learning 1 ; he was a competent mathematician ; he was 
even a poet. At the same time he was a believer in all sorts of 
myths and mysteries and a devout worshipper of divinities both 
Greek and Oriental. 

Though he was a competent mathematician, he was evidently 
much more a philosopher than a mathematician 1 . This is shown 
even in his commentary on Eucl. L, where, not only in the Prologues 
(especially the first), but also in the notes themselves, he seizes any 
opportunity for a philosophical digression. He says himself that he 
attaches most importance to "the things which require deeper study 
and contribute to the sum of philosophy 3 "; alternative proofs, cases, 
and the like (though he gives many) have no attraction for him ; 
and, in particular, he attaches no value to the addition of Heron to 
I, 47*, which is of considerable mathematical interest Though he 
esteemed mathematics highly, it was only as a handmaid to philosophy 
He quotes Plato's opinion to the effect that "mathematics, as making 
use of hypotheses, falls short of the non- hypothetical and perfect 
science 6 ". .."Let us then not say that Plato excludes mathematics 
from the sciences, but that he declares it to be secondary to the one 
supreme science'." And again, while "mathematical science must be 
considered desirable in itself, though not with reference to the needs 
of daily life," "if it is necessary to refer the benefit arising from it to 
something else, we must connect that benefit with intellectual know- 
ledge (voepitv yvaxrtv), to which it leads the way and is a propaedeutic, 
clearing the eye of the soul and taking away the impediments which 
the senses place in the way of the knowledge of universals (r&p 
$\an>y." 

We know that in the Neo-Platonic school the younger pupils 
learnt mathematics ; and it is clear that Proclus taught this subject, 
and that this was the origin of the commentary. Many passages 
show him as a master speaking to scholars. Thus "we Have illustrated 

1 Zeller calls him "Der Gelehrtc, dem kern Feld damaligen Wissens verschlossen ist." 

* Van Pesch observes that in his commentaries on the Timaeus (pp. 671 — 2) he speaks 
as no real mathematician could have spoken. In the passage referred to the question is 
whether the sun occupies a middle place among the planets. Proclus rejects the view of 
Hipparchus and Ptolemy because "& Vtwpy6t" (sc. the Chaldean, says Zeller} thinks otherwise, 
"whom it is not lawful to disbelieve-" Martin says rather neatly, " Pour Proclus, leg 
Elements d'Euclide ont l 1 hen reuse chance de n'elre contredits ni par fes Oracles chaldalques, 
ibi par les speculations des pythagoriciens anciens et nouveaux ,.,,,." 

* Proclus, p. 84, 13. ' ibid. p. 429, tit, 

6 ibid. p. 31, 20. ■ ibid. p. 31, «. 

7 ibid. p. 37, J7 to 18, 7; cf. also p, »i, 15, pp. 46, +7. 



ch. iv] PROCLUS AND HIS SOURCES 31 

and made plain all these things in the case of the first problem, but 
it is necessary that my hearers should make the same inquiry as 
regards the others as well '," and " I do not indicate these things as a 
merely incidental matter but as preparing us beforehand for the 
doctrine of the Timaeus 1 .* 1 Further, the pupils whom he was 
addressing were beginners in mathematics ; for in one place he says 
that he omits "for the present" to speak of the discoveries of those 
who employed the curves of Nicomedes and Hippias for trisecting 
an angle, and of those who used the Archimedean spiral for dividing 
an angle in any given ratio, because these things would be too 
difficult for beginners (Svff&ewpijTovs to« tla-ayoftivoi^)'. Again, if 
his pupils had not been beginners, it would not have been necessary 
for Proclus to explain what is meant by saying that sides subtend 
certain angles', the difference between adjacent and vertical angles' 
etc., or to exhort them, as he often does, to work out other particular 
cases for themselves, for practice {yvpvaaias ihteica)'. 

The commentary seems then to have been founded on Proclus' 
lectures to beginners in mathematics. But there are signs that it 
was revised and rt -edited for a larger public ; thus he gives notice in 
one place' "to those who shall come upon" his work (roh ivrev^o- 
fiivov;). There are also passages which could not have heen under- 
stood by the beginners to whom he lectured, e.g. passages about the 
cylindrical helix', conchoids and cissoids*. These passages may have 
been added in the revised edition, or, as van Pesch conjectures, the 
explanations given in the lectures may have been much fuller and 
more comprehensible to beginners, and they may haw; been shortened 
on revision. 

In his comments on the propositions of Euclid, Proclus generally 
proceeds in this way : first he gives explanations regarding Euclid's 
proofs, secondly he gives a few different cases, mainly for the sake of 
practice, and thirdly he addresses himself to refuting objections 
raised by cavillers to particular propositions. The latter class of 
note he deems necessary because of "sophistical cavils" and the 
attitude of the people who rejoiced in finding paralogisms and in 
causing annoyance to scientific men". His commentary does not 
seem to have been written for the purpose of correcting or improving 
Euclid. For there are very few passages of mathematical content 
in which Proclus can be supposed to be propounding anything of his 
own ; nearly all are taken from the works of others, mostly earlier 
commentators, so that, for the purpose of improving on or correcting 
Euclid, there was no need for his commentary at all. Indeed only in 
one place does he definitely bring forward anything of his own to get 
over a difficulty which he finds in Euclid"; this i, c where he tries to 

I Proclm, p. no, 1 9. ' il/id. p. 384, a. 

* ibid. p. 171, 11. * ibid, p. 338, 1*. 

* ibid.?. 19S, 14. * Cf. p. 114, is {on i. »). 

* ibid. p. 84, 9. ' ibid- p. 105. 

* ibid. p. 1 ti. " 'bid. p. 375, 9. 

II ibid. pp. 368—373. 



32 INTRODUCTION [ch. iv 

prove the parallel-postulate, after first giving Ptolemy's attempt and 
then pointing out objections to it. On the other hand, there are a 
number of passages in which he extols Euclid; thrice 1 also he supports 
Euclid against Apollonius where the latter had given proofs which he 
considered better than Euclid's (I. 10, II, and 23). 

Allusion must be made to the debated question whether Proclus 
continued his commentaries beyond Book I, His intention to do so 
is clear from the following passages. Just after the words above 
quoted about the trisection etc. of an angle by means of certain curves 
he says, " For we may perhaps more appropriately examine these 
things on the third book, where the writer of the Elements bisects a 
given circumference*." Again, after saying that of all parallelograms 
which have the same periifieter the square is the greatest " and the 
rhomboid least of all," he adds : "But this we will prove in another 
place ; for it is more appropriate to the (discussion of the) hypotheses 
of the second book'." Lastly, when alluding (on I. 45) to the squaring 
of the circle, and to Archimedes' proposition that any circle is equal 
to the right-angled triangle in which the perpendicular is equal to the 
radius of the circle and the base to its perimeter, he adds, "But of this 
elsewhere* " ; this may imply, an intention to treat of the subject on 
Eucl. XII., though Heiberg doubts It*. But it is clear that, at the time 
when the commentary on Book I. was written, Proclus had not yet 
begun to write on the other Books and was uncertain whether he 
would be able to do so : for at the end he says", " For my part, if I 
should be able to discuss the other books' in the same manner, I 
should give thanks to the gods ; but, if other cares should draw me 
away, I beg those who are attracted by this subject to complete the 
exposition of the other books as well, following the same method, and 
addressing themselves throughout to the deeper and better defined 
questions involved " (to irpayfunetaiBe<s iravraxov /cat evhiaLperov 
nerahttoKOvTfx;). 

There is in fact no satisfactory evidence that Proclus did actually 
write any more commentaries than that on Book I." The contrary 
view receives support from two facts pointed out by Heiberg, viz. (1) 
that the scholiast's copy of Proclus was not so much better than our 

1 Proclus, p. 1S0, 91 p. 183, 10; pp. 335, 3j6. 2 ibid, p. 373, 14. 

' ibid. p. 39.8, i8. • ibid. p. 41J, 6. 

* Heiberg, EailidStuditn, p. [65, note. * Proclus, p. 431, g, 

T The words in the Greek are : */ p£p ivvTjSditfitv tal toU \01rait tw txbrby Tpbrov iftMdr, 
For i£ek$tai Heiberg would read {*t(tKti'w. 

* True, a Vatican MS. has a collection of scholia on Books 1. (extracts from the extant 
commentary of Proclus), It., v., vt., x. headed Eij rd EvK\tL6ov aTOiXfia TpQXtLfifitw&fitfa 4k 
tuv f h V''.- .V:u.-'-!T;rn,i"-a^p irai kilt' iitiTBti.-b.v* Heiberg holds that this title itself suggests that the 
authorship of Proclus was limited to the scholia on Book I. ; for rfnAniiparifien iit Twr 
llpit\ov suits extracts from Proclus' prologues, but hardly scholia to later Books. Again, a 
certain scholium (Heiberg in jftraun. X.XXV1II., [903, p. 341, No. 17) purports to quote 
words from the end of "a scholium of Proclus" on x. 9. The words quoted are from the 
scholium x. No. 61, one of the Scholia Vaticana. But none of the other, older, sources 
connect Proclus' name with X. No. 61 ; it is probable therefore that a Byiantine, who had in 
bis Ms, of Euclid the collection of Schol- Vat, and knew that those on !><..■!, I. came from 
Proclus, himself attached Proclus' name to the others- 



ch. iv] PROCLUS AND HIS SOURCES 33 

MSS. as to suggest that the scholiast had further commentaries of 
Proclus which have vanished for us 1 ; (2; that there is no trace in the 
scholia of the notes which Proclus promised in the passages quoted 
above. 

Coming now to the question of the sources of Proclus, we may say 
that everything goes to show that his commentary is a compilation, 
though a compilation "in the better sense" of the term". He does not 
even give us to understand that we shall find in it much of his own ; 
" let us," he says, " now turn to the exposition of the theorems proved 
by Euclid, selecting the more subtle of the comments made on them 
by the ancient writers, and cutting down their interminable diffuse- 
ness...*": not a word about anything of his own. At the same time, 
he seems to imply that he will not necessarily on each occasion quote 
the source of each extract from an earlier commentary ; and, in fact, 
while he quotes the name of his authority in many places, especially 
where the subject is important, in many others, where it is equally 
certain that he is not giving anything of his own, he mentions no 
authority. Thus he quotes Heron by name six times ; but we now 
know, from the commentary of an-Nairizi, that a number of other 
passages, where he mentions no name, are taken from Heron, and 
among them the. not unimportant addition of an alternative proof to 
I. 19. Hence we can by no means conclude that, where no authority 
is mentioned, Proclus is giving notes of his own. The presumption is 
generally the other way ; and it is often possible to arrive at a con- 
clusion, either that a particular note is not Proclus' own, or that it 
is definitely attributable to someone else, by applying the ordinary 
principles of criticism. Thus, where the note shows an unmistakable 
affinity to another which Proclus definitely attributes to some com- 
mentator by name, especially when both contain some peculiar and 
distinctive idea, we cannot have much doubt in assigning both to the 
same commentator*. Again, van Pesch finds a criterion in the form 
of a note, where the explanation is so condensed as to be only just 
intelligible; the note is that in which a converse of I. 32 is proved" 
the proposition namely that a rectilineal figure which has all its in- 
terior angles together equal to two right angles is a triangle. 

It is not safe to attribute a passage to Proclus himself because he 
uses the first person in such expressions as " I say " or " I will prove " 
— for he was in the habit of putting into his own words the substance 
of notes borrowed from others — nor because, in speaking of an 

1 While one class or scnuua (Schol. Vat.) have some better readings than our MSS. of 
Proclus h ave, and partly Fill up the gaps at 1. 36, 37 and I. 41 — 43, the other class (SchgL 
Vind.) derive from an inferior Proclus MS, which also had the same lacunae. 

a Knoche, UnUrsttchungcn iibcr dti Proklui Diattoihui Commtntar zu EttktidTi Eie- 
mentm {186?), p. 1 1. 

3 Proclus, p. 30o, to — 13. 

4 Instances of the application of this criterion will be found in the discussion of Proclus' 
indebtedness to the commentaries of Heron, Porphyry and Pappus, 

* Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the 
proof is followed by a passage which, on comparison with Pappus' note on the postulate that 
all right angles are equal, he feels justified in assigning to Pappus, I doubt if the evidence is 
sufficient. 



34 INTRODUCTION [ch. iv 

objection raised to a particular proposition, he uses such expressions 
as "perhaps someone may object" («rwe 8* &v tii>« it/trraUv...): for 
sometimes other words in the same passage, indicate that the objection 
had actually been taken by someone 1 . Speaking generally, we shall 
not be justified in concluding that Prod us is stating something new of 
his own unless he indicates this himself in express terms. 

As regards the form of Proclus' references to others by name, van 
Pesch notes that he very seldom mentions the particular zvork from 
which he is borrowing. If we leave out of account the references to 
Plato's dialogues, there are only the following references to books : 
the Bacchae of Philolaus 1 , the Symmikta of Porphyry*, Archimedes On 
the Sphere and Cylinder 1 , Apollonius On the cochlias*, a book by 
Eudemus on The Angle', a whole book of Posidonius directed against 
Zeno of the Epicurean sect', Carpus' Astronomy', Eudemus' History of 
Geometry*, and a tract by Ptolemy on the parallel-postulate 10 . 

Again, Proclus does not always indicate that he is quoting some- 
thing at second-hand. He often does so, e.g. he quotes Heron as the 
authority for a statement about Phi lip pus, Eudemus as attributing a 
certain theorem to Oenopides etc. ; but he says on i. 1 2 that " Oeno- 
pides first investigated this problem, thinking it useful for astronomy " 
when he cannot have had Oenopides' work before him. 

It has been said above that Proclus was in the habit of stating in 
his own words the substance of the things which he borrowed. We 
are prepared for this when we find him stating that he will select the 
best things from ancient commentaries and "cut short their intermin- 
able diffuseness," that he will "briefly describe" (oiWTo/ttif laTopqaai) 
the other proofs of 1. 20 given by Heron and Porphyry and also the 
proofs of I, 25 by Menelaus and Heron. But the best evidence is of 
course to be found in the passages where he quotes works still extant, 
e.g. those of Plato, Aristotle and Plotinus. Examination of these 
passages shows great divergences from the original; even where he 
purports to quote textually, using the expressions " Plato says," or 
" Plotinus says," he by no means quotes word for word". In fact, he 
seems to have had a positive distaste for quoting textually from other 
works. He cannot conquer this even when quoting from Euclid ; he 
says in his note on I. 22, " we will follow the words of the geometer " 
but fails, nevertheless, to reproduce the text of Euclid unchanged". 

We now come to the sources themselves from which Proclus drew 

I Van Pesch illustrates this by ;ui objection refuted in the note on I. 9, p. 1JJ, 11 sqq. 
After using the above expression to introduce the objection, Proclus uses further on (p. »73,ij) 
the term "they say" ijnelr). 

• Proclus, p. »a, 15. * ibid, p, j6, ij. 

* ibid. p. 71, 18. * ibid. p. 105, 3. 
ibid. p. j 35, 8. T ibid. p. aoo, 3. 
■ ibid. p. j+i, 19. "' Hid. p. 351, tj. 

10 ibid. p. 361, r5, 

II See the passages referred to by van Pesch (p. 70}. The most glaring case is a passage 
(p. 31, 10) where he quotes Plotinus, using the expression " Plotinus says......" Comparison 

with Plotinus. Hnntad. t. 3, 3, shows that very few words are those of Plotinus himself; the 
rest represent Plotinus' views in Proclus' own language. 

11 Proclus, p. 330, 19 sqq 



ch. iv] PROCLUS AND HIS SOURCES 35 

in writing his commentary. Three have already been disposed of, 
viz. Heron, Porphyry and Pappus, who had all written commentaries 
on the Elements 1 . We go on to 

Eudemus, the pupil of Aristotle, who, among other works, wrote a 
history of arithmetic, a history of astronomy, and a history of geometry. 
The importance of the last mentioned work is attested by the frequent 
use made of it by ancient writers. That there was no other history 
of geometry written after the time of Eudemus seems to be proved by 
the remark of Proclus in the course of his famous summary r " Those 
who compiled histories bring the development of this science up to 
this point. Not much younger than these is Euclid*. . .." The loss of 
Eudemus' history is one of the gravest which fate has inflicted upon 
us, for it cannot be doubted that Eudemus had before htm a number 
of the actual works of earlier geometers, which, as before observed, 
seem to have vanished completely when they were superseded by the 
treatises of Euclid, Archimedes and Apoilonius. As it is, we have to 
be thankful for the fragments from Eudemus which such writers as 
Proclus have preserved to us. 

I agree with van Pesch' that there is no sufficient reason for 
doubting that the work of Eudemus was accessible to Proclus at first 
hand. For the later writers Simplicius and Eutocius refer to it in 
terms such as leave no room for doubt that tftey had it before them. 
I have already quoted a passage from Simplicius' account of the lunes 
of Hippocrates to the effect that Eudemus must be considered the 
best authority since he lived nearer the times*. In the same place 
Simplicius says", " I will set out what Eudemus says word for word 
(tcwrh. \il-tv Xeyofifva), adding only a little explanation in the shape of 
reference to Euclid's Elements owing to the memorandum-like style of 
Eudemus (St a tov iriroji.viifi.aTticbv Tpoirov rov EvSjj/*ou) who sets out 
his explanations in the abbreviated form usual with ancient writers. 
Now in the second book of the history of geometry he writes as 
follows'." It is not possible to suppose that Simplicius would have 
written in this way about the style of Eudemus if he had merely been 
copying certain passages second-hand out of some other author and 
had not the original work itself to refer to. In like manner, Eutocius 
speaks of the paralogisms handed down in connexion with the 
attempts of Hippocrates and Antiphon to square; the circle", "with 
which I imagine that those are accurately acquainted who have 
examined (^reo-Ke^tei-ow) the geometrical history of Eudemus and 
know the Ceria Aristotelica." How could the contemporaries of Euto- 
cius have examined the work of Eudemus unless it was still extant in 
his time ? 

The passages in which Proclus quotes Eudemus by name as his 
authority are as follows : 

(l) On I. 26 he says that Eudemus in his history of geometry 

1 See pp. 30 to a 7 above. 

* l*roclus, p. 68, 4—7. * De Prodi fontibui. pp. ft— Jg. 

* See above, p. 19. ' Simplicius, Jet. tit., ed. Dids, p. 60, 17. 

* Archimedes, ed. Heiberg, vol. ill. p. 11&. 



36 INTRODUCTION [ch. iv 

referred this theorem to Thales, inasmuch as it was necessary to 
Thales 1 method of ascertaining the distance of ships from the shore 1 . 

(2) Eudemus attributed to Thales the discovery of Eucl. I. 15", 
and 

(3) to Oenopides the problem of I. 23*. 

(4) Eudemus referred the discovery of the theorem in 1. 32 to the 
Pythagoreans, and gave their proof of it, which Proclus reproduces*. 

(5) On I. 44 Proclus tells us* that Eudemus says that "these 
things are ancient, being discoveries of the Pythagorean muse, the 
application (irapafioXrf) of areas, their exceeding (vTrcpfioXij) and 
their falling short (lA-Xe^ec)," The next words about the appro- 
priation of these terms (parabola, hyperbola and ellipse) by later 
writers (i.e. Apollonius) to denote the conic sections are of course not 
due to Eudemus. 

Coming now to notes where Eudemus is not named by Proclus, 
we may fairly conjecture, with van Pesch, that Eudemus was really 
the authority for the statements (1) that Thales first proved that a 
circle is bisected by its diameter* (though the proof by reductio ad 
absurdum which follows in Proclus cannot be attributed to Thales 7 ), 
(z) that " Plato made over to Leodamas the analytical method, by 
means of which it is recorded (ltrTopi)T<u) that the latter too made 
many discoveries in geometry"," (3) that the theorem of I. 5 was due 
to Thales, and that for equal angles he used the more archaic 
expression "similar" angles', (4) that Oenopides first investigated 
the problem of I. 12, and that he called the perpendicular the 
gnomonic line (icara yvo>ftova.y°, (5) that the theorem that only three 
sorts of polygons can fill up the space round a point, viz. the 
equilateral triangle, the square and the regular hexagon, was 
Pythagorean". Eudemus may also be the authority for Proclus' 
description of the two methods, referred to Plato and Pythagoras 
respectively, of forming right-angled triangles in whole numbers". 

We cannot attribute to Eudemus the beginning of the note on 
I. 47 where Proclus says that "if we listen to those who like to 
recount ancient history, we may find some of them referring this 
theorem to Pythagoras and saying that he sacrificed an ox in honour 
of his discovery"." As such a sacrifice was contrary to the Pytha- 
gorean tenets, and Eudemus could not have been unaware of this, 
the story cannot rest on his authority. Moreover Proclus speaks as 
though he were not certain of the correctness of the tradition ; indeed, 

1 Proclus, p. 331, 14 — 18. ■ ibid, p. 199, 3. 

* ibid. p. 333, 5. * Hid. p. 379, 1—16. 

* Hid. p. + iy, ij — 18. * ibid. p. 157, 10, 11. 

7 Cantor [GcscA, d. Math. Is, p. ait) points out the connexion between the rrducth ad 
absurdum And the Analytical method said to have been discovered by Plato, Proclus gives 
the proof by rfduclio ad absurdum to meet an imaginary critic who desires a mathematical 
proof ; possibly Thales may have been satisfied with the argument in the same sentence 
which mentions Thales, "the cause o r the bisection being the unswerving course of the 
straight line through the centre." 

* Proclus, p. 31 1, 19 — 13. * ibid. p. 350, 10. 

" ibid. p. 383, ; — 10. '* Vrid. pp. 304, 11 — 303, 3. 

u ibid. pp. 418, 7 — 419, 9. a Hid. p. 416, 6 — 9. 



ch. iv] PROCLUS AND HIS SOURCES 37 

so far as the story of the sacrifice is concerned, the same thing is told 
of Thales in connexion with his discovery that the angle in a semi- 
circle is a right angle 1 , and Plutarch is not certain whether the ox 
was sacrificed on the discovery of I. 47 or of the problem about 
application of areas*. Plutarch's doubt suggests that he knew of no 
evidence for the story beyond the vague allusion in the distich of 
Apollodorus " Logisticus " (the " calculator ") cited by Diogenes 
Laertius also'; and Proclus may have had in mind this couplet with 
the passages of Plutarch. 

We come now to the question of the famous historical summary 
given by Proclus*. No one appears to maintain that Eudemus is the 
author of even the early part of this summary in the form in which 
Proclus gives it. It is, as is well known, divided into two distinct 
parts, between which comes the remark, " Those who compiled 
histories* bring the development of this science up to this point. 
Not much younger than these is Euclid, who put together the 
Elements, collecting many of the theorems of Eudoxus, perfecting 
many others by Theaetetus, and bringing to irrefragable demonstration 
the things which had only been somewhat loosely proved by his pre- 
decessors." Since Euclid was later than Eudemus, it is impossible that 
Eudemus can have written this. Yet the style of the summary after 
this point does not show any such change from that of the former 
portion as to suggest different authorship. The author of the earlier 
portion recurs frequently to the question of the origin of the 
elements of geometry in a way in which no one would be likely to 
do who was not later than Euclid ; and it must be the same hand 
which in the second portion connects Euclid's Elements with the 
work of Eudoxus and Theaetetus'. 

If then the summary is the work of one author, and that author 
not Eudemus, who is it likely to have been ? Tannery answers that 
it is Geminus 1 ; but I think, with van Pesch, that he has failed to 
show why it should be Geminus rather than another. And certainly 
the extracts which we have from Geminus' work suggest that the sort 
of topics which it dealt with was quite different ; they seem rather to 
have been general questions of the content of mathematics, and even 
Tannery admits that historical details could only have come inci- 
dentally into the work'. 

Could the author have been Proclus himself? Circumstances 

1 Diogenes Laertius, I. 14, p. 6, ed. Cobet. 

1 Plutarch, nen posse tttaviter vivi secundum Epicurum, 1 1 ; Symp- VIM, 1. 

■ Ding, Laert. vm. is, p. 107, ed. Cobet: 

'livipta Hv0ay6pnt ri refHK\tit fGpcTo fp&fifia, 
Kfii 1$ Sertfi x\uvi)v ifyaye @w0 vv^v. 
See on this subject Tannery, La Giomttrii gr&cqu4 t p. 105. 

* Proclus, pp. 64 — 70. 

* The plural is well explained by Tannery, La Giomitrii gricqut t pp. 73, 74. No doubt 
the author of the summary tried to supplement Eudemus by means of any other histories 
which threw light on the subject. Thus e.g. the allusion {p. 64, 11) to the Nile recalls 
Herodotus. Cf. the depression in Proclus, p. 64, 19, rapi Tar roXkar iirripijTiu. 

* Tannery, La Gimiitrit grtequt, p. ;j. 

' Hid. pp. 66 — 7 j. ' ibid. p. in. 



38 INTRODUCTION [ch, iv 

which seem to suggest this possibility are (i) that, as already stated, 
the question of the origin of the Elements is kept prominent, 
(2) that there is no mention of Democritus, whom Eudemus would 
not be likely to have ignored, while a fol tower of Plato would be 
likely enough to do him the injustice, following the example of Plato 
who was an opponent of Democritus, never once mentions him, and 
is said to have wished to burn all his writings 1 , and (3) the allusion at 
the beginning to the "inspired Aristotle" (0 Baifiovux; 'Apto-ToTeXi;?)*, 
though this may easily have been inserted by Proclus in a quotation 
made by him from someone else. On the other hand there are 
considerations which suggest that Proclus himself was not the writer. 
(1) The style of the whole passage is rtot such as to point to him 
as the author. (2) If he wrote it, it is hardly conceivable that he 
would have passed over in silence the discovery of the analytical 
method, the invention of Plato to which he attached so much 
importance*. 

There is nothing improbable in the conjecture that Proclus quoted 
the summary from a compendium of Eudemus' history made by some 
later writer: but as yet the question has not been definitely settled. 
All that is certain is that the early part of the summary must have 
been made up from scattered notices found in the great work of 
Eudemus. 

Proclus refers to another work of Eudemus besides the history, 
viz. a book on The Angle [0t0Xlov wepl ya>vla<;)*. Tannery assumes 
that this must have been part of the history, and uses this assumption 
to confirm his idea that the history was arranged according to subjects, 
not according to chronological order*. The phraseology of Proclus 
however unmistakably suggests a separate work ; and that the 
history was chronologically arranged seems to be clearly indicated by 
the remark of Simplicius that Eudemus "also counted Hippocrates 
among the more ancient writers " {iv tok vd\atoT4poi<;) a . 

The passage of Simplicius about the lunes of Hippocrates throws 
considerable light on the style of Eudemus' history. Eudemus wrote 
in a memorandum-like or summary manner {jov vTrofivrifiaTiKov rpoirov 
tow EuS?jjM>ii)' when reproducing what he found in the ancient writers; 
sometimes it is clear that he left out altogether proofs or constructions 
of things by no means easy'. 

Gem in u 9. 

The discussions about the date and birthplace of Geminus form a 
whole literature, as to which 1 must refer the reader to Manitius and 
Tittel', Though the name looks like a Latin name (Gemlnus), Mani- 

1 Diog. Laertius, Ix. 40, p. 337, ltd. Cobet. - Proclus, p. 64, 8. 

* Proclus, p. in, [9 sqq. j the passage is quoted above, p. 36. 

* ibid, p. 125, 8. i Tannery, La Giotnltrit grettjite, p. 26. 

* Simplicius, «d. Diels, p. 60, 23. ' ibid, p. - 60, 29, 

* Cf. Simplicius, p. 63, 19 sqq. ; p. 64. 15 sqo. ; also Usener's note *'dc supplcndis 
Hippocratis quas omisit Eudemus cons tructioni bus added to Diels' preface, pp. xxiii — xxvi. 

'Manitius, Gemini ticmenia. astronotnitM (Tcubner, 1898), pp. 237- — 151; Tittel, ait. 
*' Geminos " in Pauly-Wissowa's Rtal-Eiuytiopiidif. dtr ttassischm Aitertutnswisiemehaft, 
vol. Vlt.. 1910. 



ch. tv] PROCLUS AND HIS SOURCES 39 

tius concluded that, since it appears as VeftZvos in all Greek MSS. and 
as rf^Ktiio? in some inscriptions, it is Greek and possibly formed from 
7e/A as 'Epyiuo? is from ipy and *AX.ef wot from aXef (cf. also 'I*tm'os, 
KpaTivos). Tittel is equally positive that it is Gemtnus and suggests 
that Te/iifo? is due to a false analogy with 'AXeftetx? etc. and Fe/tetpoc 
wrongly formed on the model of 'Avravuvo?, ' kypwrreiva. Geminus, 
a Stoic philosopher, born probably in the island of Rhodes, was the 
author of a comprehensive work on the classification of mathematics, 
and also wrote, about 73-67 B.C., a not !ess comprehensive commentary 
on the meteorological textbook of his teacher Pnsidonius of Rhodes. 

It is the former work in which we are specially interested here. 
Though Proclus made great use of it, he does not mention its title, 
unless we may suppose that, in the passage (p. 177, 24) where, after 
quoting from Geminus a classification of lines which never meet, he 
says, " these remarks I have selected from the tyXoKoKia of Geminus," 
(ptXoxaXla is a title or an alternative title. Pappus however quotes a 
work of Geminus "on the classification of the mathematics" (iv t$ 
wept Tt}<; t&v naffyfidrtov rafetus-) 1 , while Eutocius quotes from *' the 
sixth book of the doctrine of the mathematics" (tv ra> eVro) 1770 t&v 
fiaffiHidraiv QtwpiasV Tannery' pointed out that the former title 
corresponds well, enough to the long extract* which Proclus gives in 
his first prologue, and also to the fragments contained in the Anonymi 
variae collectiones published by Hultsch at the end of his edition of 
Heron'; but it does not suit most of the o'ther passages borrowed by 
Proclus, The correct title was therefore probably that given by 
Eutocius, The Doctrine, or Theory, of the Mathematics ; and Pappus 
probably refers to one particular portion of the work, say the first 
Book. If the sixth Book treated of conies, as we may conclude from 
Eutocius, there must have been more Books to follow, because Proclus 
has preserved us details about higher curves, which must have come 
later. If again Geminus finished his work and wrote with the same 
fulness about the other branches of mathematics as he did about 
geometry, there must have been a considerable number of Books 
altogether. At all events it seems to have been designed to give 
a complete view of the whole science of mathematics, and in fact to 
be a sort of encyclopaedia of the subject. 

I shall now indicate first the certain, and secondly the probable, 
obligations of Proclus to Geminus, in which task 1 have only to follow 
van Pesch, who has embodied the results of Tittel's similar inquiry also*. 
I shall only omit the passages as regards which a case for attributing 
them to Geminus does not seem to me to have been made out. 

First come the following passages which must be attributed to 
Geminus, because Proclus mentions his name: 

(1) (In the first prologue of Proclus') on the division of mathe- 

1 Pappus, ed, Hultsch, p. toi6, 9. ' Apollonius, fid. Heiberg, vol. II. p. 170. 

* Tannery, La Giomttru grttiiut, pp. 18, 19. * Proclus, pp. 38, I — 41, 8. 

* Heron, ed. Hultsch, pp. 140, 16—149, li - 

•Van Pesch, Di Prgck fonlibus, pp. 97—113. The dissertation of Tittel is entitled Dt 
Gtmini Steici ttudiit maihrmatuis ( 1895}- 

' Proclus, pp, 38, I — 41, 8, except the allusion in p. 41, S — 10, to Ctesibius and Heron and 



4 o INTRODUCTION [ch. iv 

matical sciences into arithmetic, geometry, mechanics, astronomy, 
optics, geodesy, canonic (science of musical harmony), and logistic 
(apparently arithmetical problems); 

(3) (in the note on the definition of a straight line) on the 
classification of lines (including curves) as simple (straight or circular) 
and mixed, composite and incomposite, uniform (ou-oiofiepels;) and 
non-uniform (avoitotoptpeis), lines " about solids " and lines produced 
by cutting solids, including conic and spiric sections' ; 

(3) (in the note on the definition of a plane surface) on similar 
distinctions extended to surfaces and solids' ; 

(4) (in the note on the definition of parallels) on lines which 
do not meet (d<rt!^wro»Toj) but which are not on that account 
parallel, e.g. a curve and its asymptote, showing that the property of 
not meeting does not make lines parallel — a favourite observation of 
Geminus — and, incidentally, on bounded lines or those which enclose a 
figure and those which do not* ; 

(5) (in the same note) the definition of parallels given by 
Posidonius' ; 

(6) on the distinction between postulates and axioms, the futility 
of trying to prove axioms, as Apollonius tried to prove Axiom 1 , and 
the equal incorrectness of assuming what really requires proof, " as 
Euclid did in the fourth postulate [equality of right angles] and in 
the fifth postulate [the parallel-postulate]*" ; 

(7) on Postulates 1, 2, 3, which Geminus makes depend on the 
idea of a straight line being described by the motion of a point* ; 

(8) (in the note on Postulate 5) on the inadmissibility in geometry 
of an argument which is merely plausible, and the danger in this 
particular case owing to the existence of lines which do converge 
ad infinitum and yet never meet 7 ; 

(9) (in the note on 1. 1) on the subject-matter of geometry, 
theorems, problems and Btopurftoi (conditions of possibility) for 
problems' ; 

(10) (in the note on 1, 5) on a generalisation of I. 5 by Geminus 
through the substitution for the rectilineal base of " one uniform line 
(curve)," by means of which he proved that the only " uniform lines " 

their pneumatic devices (fav/wrersiVinf}, as regards which ['roc! us" authority may be Pappus 
(ytii. p. 1014, 14 — ay) who uses very similar expressions. Heron, even if not later than 
Geminus, could hardly have been included in a historical work by him. Perhaps Geminus 
may have referred to Ctesibius only, and Proclus may have inserted " and Heron himself. 



1 Proclus, pp. 103, 11 — 107, 10; pp. nt, 1 — 113, 3. 
* iiid. pp. 117, 14 — 110, 11, where perhaps in the pa 



.pp. 117, 14 — 110, it, where perhaps in the passage pp. 117, in — ti8, 1$ we may 
have Geminus' own words. 

* iiid. pp. 176, 18—177, ij; perhaps also p. 175. The note ends with the words 
"These things too we have selecled from Geminus' *iX«aWa for the elucidation of the 
matters in question." Tannery (p. 17) takes these words coming at the end of the commen- 
tary on the definitions as referring to the whole of the portion of Ihe commentary dealing 
with the definitions. Van Pesch properly regards them as only applying to the note on 
paralltli. This seems to me clear from the use of the word loo (TanSm mi). 

* Proclus, p. 176, 5 — 17. 

* ibid. pp. 17a — 181, 4; pp. 183, 14— 184, 10; cf. p. 188, 3— II. 

* ibid, p. 185, 6 — 15. 

' iiid. p. 10.1, j — 19. ' ibid. pp. too, it — 101, 15. 



CH. iv] PROCXUS AND HIS SOURCES 41 

(alike in all their parts) are a straight line, a circle, and a cylindrical 
helix'; 

(11) (in the note on I. 10) on the question whether a line is made 
up of indivisible parts (dfttpij), as affecting the problem of bisecting 
a given straight line 1 ; 

(12) (in the note on 1. 35) on (apical, or /o««-theorems', where 
the illustration of the equal parallelograms described between a 
hyperbola and its asymptotes may also be due to Geminus*. 

Other passages which may fairly be attributed to Gem in us, though 
his name is not mentioned, are the following : 

(1) in the prologue, where there is the same allusion as in the 
passage (8) above to a remark of Aristotle that it is equally absurd to 
expect scientific proofs from a rhetorician and to accept mere plausi- 
bilities from a geometer' ; 

(2) a passage in the prologue about the subject-matter, methods, 
and bases of geometry, the latter including axioms and postulates*; 

(3) another on the definition and nature of elements 1 ; 

(4) a remark on the Stoic use of the term axiom for every simple 
statement (dirotfxtvirtv <z7r\i)) s ; 

(5) another discussion on theorems and problems 6 , in the middle 
of which however there are some sentences by Prod us himself 1 '. 

(6) another passage, in connexion with Def. 3, on lines including 
or not including a figure (with which cf. part of the passage (4) 
above) 11 ; 

(7) a classification of different sorts of angles according as they 
are contained by simple or mixed lines (or curves)" ; 

(8) a similar classification of figures", and of plane figures'*; 

(9) Posidonius' definition of a figure 1 * ; 

(10) a classification of triangles into seven kinds" ; 

(11) a note distinguishing lines (or curves) producible indefinitely 
or not so producible, whether forming a figure or not forming a 
figure (like the "single-turn spiral") 17 ; 

( 1 2) passages distinguishing different sorts of problems", different 
sorts of theorems", and two sorts of converses (complete and partial)*; 

(13) the definition of the term " porism " as used in the title of 
Euclid's Porisms, as distinct from the other meaning of " corollary ""; 

(14) a note on the Epicurean objection to I. 20 as being obvious 
even to an ass 11 ; 

(15) a passage on the properties of parallels, with allusions to 

I Proclus, p. aji, 1 — 11. ■ Hid. pp. 177, 15 — 179, n. 
* itid, pp. 394, 11— 39s, » and p. 39S, 13 — it. * Hid. p. 395, 8— 11. 
" »*«/- PP- 33. M— J*. I. l itid. pp, 57, 9— 58, 3. 

7 itid. pp, 71, 3—75, 4. » ibid. p. 77, 3—6. 

■ ibid. pp. 77, 7—78, 13, and 79, 3—81,4- l0 Hid. pp. 78, '3—79. «■ 

II Hid. pp. 101,-11 — 103, 18. ls Hid. pp. 116, 7 — 117, 16. 
» itid. pp. 159, 11— 160, 9. 14 Hid. pp. [6*, 17— 164, 6. 
» itid. p. 143, 5—11. " itid. p. 168, +— 11. 

17 itid. p. 187, 19— 17. " Hid. pp. j jo, 7— in. 14; also p. 330, 6—9. 

>* itid. pp. 144, 14—1+6, 11. ' » itid. pp. 151, 5— 134, 10. 

" itid. pp. 301, 11—301, 13. " itid. pp. 311, 4—313. 3' 



4» INTRODUCTION [ch. iv 

Apollonius' Conies, and the curves invented by Nicomedes, Hippias 
and Perseus 1 ; 

(16) a passage on the parallel-postulate regarded as the converse 
of I. i7«. 

Of the authors to whom P roc! us was indebted in a less degree the 
most important is Apollon ius of Perga. Two passages allude to his 
Conies*, one to a work on irrationals 1 , and two to a treatise On the 
cocklias (apparently the cylindrical helix) by Apollonius*. But more 
important for our purpose are six references to Apollonius in connexion 
with elementary geometry, 

(i) He appears as the author of an attempt to explain the idea 
of a line (possessing length but no breadth) by reference to daily 
experience, e.g. when we tell someone to measure, merely, the length 
of a road or of a wall'; and doubtless the similar passage showing 
how we may in like manner get a notion of a surface (without depth) 
is his also'. 

(2) He gave a new general definition of an angle*. 

(3) He tried to prove certain axioms', and Proclus gives his 
attempt to prove Axiom I, word for word". 

Proclus further quotes : 

(4) Apollonius' solution of the problem in Eucl. I. 10, avoiding 
Euclid's use of r, 9", 

(5) his solution of the problem in I. 11, differing only slightly 
from Euclid's", and 

(6) his solution of the problem in 1. 23 1 *, 

Heiberg" conjectures that Apollonius departed from Euclid's 
method in these propositions because he objected to solving problems 
of a more general, by means of problems of a more particular, 
character. Proclus however considers all three solutions inferior to 
Euclid's ; and his remarks on Apollonius' handling of these ele- 
mentary matters generally suggest that he was nettled by criticisms 
of Euclid in the work containing the things which he quotes from 
Apollonius, just as we conclude that Pappus was offended by the 
remarks of Apollonius about Euclid's incomplete treatment of the 
" three- and four-line locus 1 *." If this was the case, Proclus can hardly 
have got his information about these things at second-hand ; and 
there seems to be no reason to doubt that he had the actual work of 
Apollonius before him. This work may have been the treatise 
mentioned by Marinus in the words "Apollonius in his general 
treatise" CAirokXaivios iv r$ KaBoKov irpar/fitLTeiq) 1 '. If the notice 
in the Fihrist" stating, on the authority of T ha bit b. Qurra, that 

I Prdclus, pp. 355, 10—356, 16. ' ibid. p. 364, 9—11 ; pp. 36+, 10—365, 4. 

* Hid. p. 71, ig; p. 356, 8, 6. * ibid, p. 74, 13, 14. 

* ibid, pp. 105, s, 6, 14, 15. s Hid. p. too, 5—19. 

' ibid. p. 1 [4, 10 — 13. * ibid. p. 113, I J— 19 (cf. p. 114, 17. p. t»5, ij). 

* ibid. p. 183, 13, 14. 10 ibid. pp. 194, 13 — J95, 5. 

II ibid. pp. »79, 16 — 180, 4. " ibid. p. 181, 8 — 19. 

18 ibid. pp. 335, 16—336, 5. » PkiMegas, vol. xliji. p. 489. 

15 See above, pp. 1, 3. '* Marinus in Euclidis Data, ed. Menge, p. 134, 16. 

17 FiArist, U. Suter, p. 19. 



ch. iv] PROCLUS AND HIS SOURCES 43 

Apollonius wrote a tract on the parallel -postulate be correct, it may 
have been included in the same work. We may conclude generally 
that, in it, Apollonius tried to remodel the beginnings of geometry, 
reducing the number of axioms, appealing, in his definitions of lines, 
surfaces etc., more to experience than to abstract reason, and 
substituting for certain proofs others of a more general character. 

The probabilities are that, in quoting from the tract of Ptolemy in 
which he tried to prove the para I lei -postulate, Proclus had the actual 
work before him. For, after an allusion to it as "a certain book 1 " 
he gives two long extracts', and at the beginning of the second 
indicates the title of the tract, "in the (book) about the meeting of 
straight lines produced from (angles) less than two right angles," as 
he has very rarely done in other cases. 

Certain things from Posidonius are evidently quoted at second- 
hand, the authority being Geminus (e.g. the definitions of figure and 
parallels) ; but besides these we have quotations from a separate work 
which he wrote to controvert Zeno of Sidon, an Epicurean who had 
sought to destroy the whole of geometry*. We are told that Ze.10 
had argued that, even if we admit the fundamental principles (tipx<U) 
of geometry, the deductions from them cannot be proved without the 
admission of something else as well, which has not been included in 
the said principles*. On I. 1 Proclus gives at some length the argu- 
ments of Zeno and the reply of Posidonius as regards this proposition'. 
In this case Zeno's "something else" which he considers to be 
assumed is the fact that two straight lines cannot have a common 
segment, and then, as regards the " proof " of it by means of the 
bisection of a circle by its diameter, he objects that it has been 
assumed that two circumferences (arcs) of circles cannot have a 
common part. Lastly, he makes up, for the purpose of attacking it, 
another supposed " proof" of the fact that two straight lines cannot 
have a common part. Proclus appears, more than once, to be quoting 
the actual words of Zeno and Posidonius ; in particular, two expres- 
sions used by Posidonius about " the acrid Epicurean " (tov bptfivv 
'Eirucovpciav)* and his "misrepresentations" (Ho<ret8<bvi6<; ty)<rf rbv 
Ztjcowa <ruKo<f>avT€lvY. It is not necessary to suppose that Proclus 
had the original work of Zeno before him, because Zeno's arguments 
may easily have been got from Posidonius' reply ; but he would 
appear to have quoted direct from the latter at all events. 

The work of Carpus tnec/tankus (a treatise on astronomy) quoted 
from by Proclus" must have been accessible to him at first-hand, 
because a portion of the extract from it about the relation of theorems 
and problems" is reproduced word for word. Moreover, if he were not 
using the book itself, Proclus would hardly be in a position to question 
whether the introduction of the subject of theorems and problems 

' Proclus, p. 191, 13. ' ibid. pp. 363, 14— 363, IB; pp. 36s, 7—3671 *?■ 

' ibid, p. loo, 1—3. * ibid. pp. 199, u— 200, 1. 

* ibid. pp. »i4, [8 — iif, 13 [ pp. 516, 10 — 118, M. 

4 ibid. p. 3l6, 11. 7 ibid. p. ?]8, I. 

* ibid, pp. J41, 19 — »43, 11. ' Md, pp. 141, n — 143, 11. 



44 INTRODUCTION [ch. iv 

was opportune in the place where it was I'ound («* piv Kara Kaipiv rj 
^»i, Trapela0a> wpos to irapov) 1 . 

It is of course evident that Proclus had before him the original 
works of Piato, Aristotle, Archimedes and Plotinus, as well as the 
Ivfifiutrd of Porphyry and the works of his master Syrianus (d ^ftirepo^ 
Ka8fjyefta)v) % , from whom he quotes in his note on the definition of an 
angle. Tannery also points out that he must have had before him a 
group of works representing the Pythagorean tradition on its mystic, 
as distinct from its mathematical, side, from Philolaus downwards, and 
comprising the more or less apocryphal Upfc Xoyai of Pythagoras, the 
Oracles (\6yta), and Orphic verses'. 

Besides quotations from writers whom we can identify with more 
or less certainty, there are many other passages which are doubtless 
quoted from other commentators whose names we do not know. A 
list of such passages is given by van Pesch*, and there is no need to 
cite them here. 

Van Pesch also gives at the end of his work" a convenient list of 
the books which, as the result of his investigation, he deems to have 
been accessible to and directly used by Proclus, The list is worth 
giving here, on the same ground of convenience. It is as follows: 

Eudemus : history of geometry. 

Gem in us : the theory of the mathematical sciences. 

Heron : commentary on the Elements of Euclid. 

Porphyry: „ „ 

Pappus t „ „ „ 

Apollonius of Perga : a work relating to elementary geometry. 

Ptolemy : on the parallel-postulate. 

Posidonius : a book controverting Zeno of Si don. 

Carpus : astronomy. 

Syrianus j a discussion on the angle. 

Pythagorean philosophical tradition. 

Plato's works. 

Aristotle's works. 

Archimedes' works, 

Plotinus : Enneaaes. 
Lastly we come to the question what passages, if any, in the 
commentary of Proclus represent his own contributions to the subject 
As we have seen, the onus probandi must be held to rest upon him 
who shall maintain that a particular note is original on the part of 
Proclus, Hence it is not enough that it should be impossible to point 
to another writer as the probable source of a note ; we must have a 
positive reason for attributing it to Proclus, The criterion must there- 
fore be found either (i) in the general terms in which Proclus points 
out the deficiencies in previous commentaries and indicates the 
respects in which his own will differ from them, or (2) in specific 
expressions used by him in introducing particular notes which may 

1 Proclus, p. 741, j 1, 11. ■ ibid. p. 113, 19, 

8 Tannery, La GionUtrie grecque, pp. 15, 16, 

* Van Pesch, Dt Prvcli fetttitrus, p. 135. * ibid. p. 155, 



CH. iv] PROCLUS AND HIS SOURCES 45 

indicate that he is giving his own views. Besides indicating that he 
paid more attention than his predecessors to questions requiring 
deeper study (to irpayuarettSSes) and " pursued clear distinctions ' 
{to evStaiperav (ieTa8ia>Kovra<:) x — by which he appears to imply that 
his predecessors had confused the different departments of their 
commentaries, viz. lemmas, cases, and objections (eyo-rrwret?)* — Proclus 
complains that the earlier commentators had failed to indicate the 
ultimate grounds or muses of propositions*. Although it is from 
Geminus that he borrowed a passage maintaining that it is one of the 
proper functions of geometry to inquire into causes (tijii alrtav koX 
to S*a Tt)*, yet it is not likely that Geminus dealt with Euclid's 
propositions one by one ; and consequently, when we find Proclus, on 
I. 8, 16, 17, 18, 32, and 47', endeavouring to explain causes, we have 
good reason to suppose that the explanations are his own. 

Again, his remarks on certain things which he quotes from Pappus 
can scarcely be due to anyone else, since Pappus is the latest of the 
commentators whose works he appears to have used. Under this 
head, come 

(1) his objections to certain new axioms introduced by Pappus*, 

(2) his conjecture as to how Pappus came to think of his alterna- 
tive proof of I. 5', 

(3) an addition to Pappus' remarks about the curvilineal angle 
which is equal to a right angle without being one*. 

The defence of Geminus against Carpus, who combated his view 
of theorems and problems, is also probably due to Proclus', as well as 
an observation on t. 38 to the effect that I. 35 — 38 are really compre- 
hended in VI. 1 as particular cases". 

Lastly, we can have no hesitation in attributing to Proclus himself 
(1) the criticism of Ptolemy's attempt to prove the parallel-postulate", 
and (2) the other attempted proof given in the same note" (on I. 29) 
and assuming as an axiom that " if from one point two straight lines 
forming an angle be produced ad infinitum the distance between them 
when so produced ad infinitum exceeds any finite magnitude (i.e. 
length)," an assumption which purports to be the equivalent of a 
statement in Aristotle 1 *. It is introduced by words in which the 
writer appears to claim originality for his proof: "To him who 
desires to see this proved (Ka-Ta&Kevatyfieiiov) let it be said by us 
(Xeyia-ff& trap thiwv)" etc." Moreover, Philoponus, in a note on 
Aristotle's Anal. pest. 1. to, says that " the geometer (Euclid) assumes 
this as an axiom, but it wants a great deal of proof, insomuch that 
both Ptolemy and Proclus wrote a whole book upon it"." 

1 Proclui, p. 84, 1 j, p. 431, ff, j j. ■ Cf. ibid, p. *8g, u— ij ; p. 431, 13— ij. 



ibid. p. 431, 17. * ibid, p. 103, y—^$. 

Proclus, p. 

II pp. 316, T+- _ 

Proclus, p. 198, 5 — 15. 1 ibid, p. 150, 11—19. * ibid. p. 190, 9 — ^13. 



* See Proclus, p. 170, 3—14 (1. 8); pp. 309, 3—310, 8 (1. 16); pp. 310, 19—311, 33 
•T)( PP' 3'<>. '+—318, » ('• '8); p. 384, 13— j 1 (1. 31) s pp. +16, 11— 417. 8 ft. 47). 



* ibid, p. 343, 11 — 19. * ibid. pp. 403, 6—406, 9. 

11 ibid. p. 368, 1— 13. u ibid. pp. 371, 11—373, »■ 

u Aristotle, de caila, 1. 3 (171 b 18 — 30), " Produs, p. 371, 10. 

u Berlin Aristotle, voL IV. p. 114 a 9 — 11. 









CHAPTER V. 

THE TEXT 1 . 

It is well known that the title of Simson's edition of Euclid (first 
brought out in 'Latin and English in 1756) claims that, in it, "the 
errors by which Theon, or others, have long ago vitiated these books 
are corrected, and some of Euclid's demonstrations are restored " ; and 
readers of Simson's notes are familiar with the phrases used, where 
anything in the text does not seem to him satisfactory, to the effect 
that the demonstration has been spoiled, or things have been interpo- 
lated or omitted, by Theon "or some other unskilful editor." Now 
most of the MSS. of the Greek text prove by their titles that they 
proceed from the recension of the Elements by Theon ; they purport 
to be either " from the edition of Theon "(«tij? ©eWoe e'wSoVew?) or 
" from the lectures of Theon " (Avo avvovaw&v tov QsWoc). This was 
Theon of Alexandria {4th c. A.D.) who also wrote a commentary on 
Ptolemy, in which there occurs a passage of the greatest importance 
in this connexion*: "But that sectors in equal circles are to one 
another as the angles on which they stand has been proved by me in 
my edition of the Elements at the end of the sixth book." Thus Theon 
himself says that he edited the Elements and also that the second part 
of VI. 33, found in nearly all the MSS., is his addition. 

This passage is the key to the whole question of Th eon's changes 
in the text of Euclid ; for, when Peyrard found in the Vatican the 
MS. 190 which contained neither the words from the titles of the other 
MSS. quoted above nor the interpolated second part of VI. 33, he was 
justified in concluding, as he did, that in the Vatican MS. we have an 
edition more ancient than Theon 's. It is also clear that the copyist 
of P, or rather of its archetype, had before him the two recensions and 
systematically gave the preference to the earlier one ; for at xiii. 6 in 
P the first hand has added a note in the margin : " This theorem is 
not given in most copies of the new edition, but is found in those of 
the old." Thus we are more fortunate than Simson, since our 
judgment of Theon's recension can be formed on the basis, not of 
mere conjecture, but of the documentary evidence afforded by a 
comparison of the Vatican MS. just mentioned with what we may 
conveniently call, after Heiberg, the Theonine MSS. 

1 The material for the whole of this chapter is taken from Hei berg's edition of the 
Elements, introduction to vol. v., and from the same scholar's Littcrargcscktihlltihc Siudien 
iibcr Euilid, p. tJ4sqq. and ParaHpomena zit EuMidm Her-mcs, XXXVltt., too}. 

9 1. p. J 01 ed. Ililmj = |i. 50 ed. Basel. 



ch. v] THE TEXT 47 

The MSS. used for Hei berg's edition of the Elements are the 
following : 

(i) P = Vatican MS. numbered 190, 4to, in two volumes (doubt- 
less one originally) ; 10th c. 

This is the MS. which Peyrard was able to use ; it was sent from 
Rome to Paris for his use and bears the stamp of the Paris Imperial 
Library on the last page. It is well and carefully written. There are 
corrections some of which are by the original hand, but generally in 
paler ink, others, still pretty old, by several different hands, or by one 
hand with different ink in different places (P m. 2), and others again 
by the latest hand (P m. rec). It contains, first, the Elements I. — xin. 
with scholia, then Marinus' commentary on the Data (without the 
name of the author), followed by the Data itself and scholia, then the 
Elements XIV., XV. (so called), and lastly three books and a part of a 
fourth of a commentary by Theon eis rot)* irpoxeipow Kavovat IlToXe- 
fiaiou. 

The other MSS. are " Theon ine." 

(2) F = MS. xxvili, 3, in the Laurentian Library at Florence, 4to; 

10th c. 
This MS. is written in a beautiful and scholarly hand and contains 
the Elements I. — XV., the Optics and the Phaenotnena, but is not well 
preserved. Not only is the original writing renewed in many places, 
where it had become faint, by a later hand of the 1 6th c, but the same 
hand has filled certain smaller lacunae by gumming on to torn 
pages new pieces of parchment, and has replaced bodily certain 
portions of the MS., which had doubtless become illegible, by fresh 
leaves. The larger gaps so made good extend from Eucl. Vlt. 1 2 to 
IX. 1 5, and from xil. 3 to the end ; so that, besides the conclusion of the 
Elements, the Optics and Phaenomena are also in the later hand, and we 
cannot even tell what in addition to the Elements 1. — xin. the original 
MS. contained. Hei berg denotes the later hand by <f> and observes 
that, while in restoring wo;ds which had become faint and filling up 
minor lacunae the writer used no other MS., yet in the two larger 
restorations he used the Laurentian MS. XXVIII, 6, belonging to the 
13th — 14th c. The latter MS. (which Heiberg denotes by f) was 
copied from the Viennese MS. (V) to be described below. 

(3) B m Bodleian MS., D'Orville X. 1 inf. 2, 30, 4to ; A.D. 888. 
This MS. contains the Elements I. — XV. with many scholia. Leaves 

15 — 118 contain 1. 14 (from about the middle of the proposition) to 
the end of Book VI., and leaves 123 — 387 (wrongly numbered 397) 
Books VII, — XV, in one and the same elegant hand (9th c). The 
leaves preceding leaf 15 seem to have been lost at some time, leaves 
6 to 14 (containing Elem. L to the place in I. 14 above referred to) 
being carelessly written by a later hand on thick and common parch- 
ment (13th c). On leaves 2 to 4 and 122 are certain notes in the 
hand of Arethas, who also wrote a two-line epigram on leaf 5, the 
greater part of the scholia in uncial letters, a few notes and corrections, 
and two sentences on the last leaf, the first of which states that the 
MS. was written by one Stephen cleric us in the year of the world 6397 



48 INTRODUCTION [ch. v 

(= 888 a.D,), while the second records Arethas' own acquisition of it. 
Arethas lived from, say, 865 to 939 a.D. He was Archbishop of 
Caesarea and wrote a commentary on the Apocalypse. The portions 
of his library which survive are of the greatest interest to palaeography 
on account of his exact notes of dates, names of copyists, prices of 
parchment etc. It is to him also that we owe the famous Plato MS. 
from Fatmos (Cod. Clarkianus) which was written for him in November 
895 ! . 

(4) V - Viennese MS. Philos. Gr. No. 103 ; probably 12th c. 

This MS. contains 292 leaves, Eucl, Elements I. — XV. occupying 
leaves 1 to 254, after which come the Optics (to leaf 271), the 
P%aenomena (mutilated at the end) from leaf 372 to leaf 282, and lastly 
scholia, on leaves 283 to 292, also imperfect at the end. The different 
material used for different parts and the varieties of handwriting make 
it necessary for Heiberg to discuss this ms. at some length'. The 
handwriting on leaves 1 to 183 (Book 1. to the middle of X. 105) and 
on leaves 203 to 234 (from XI 31, towards the end of the proposition, 
to XIII. 7, a few lines down) is the same; between leaves 184 and 202 
there are two varieties of handwriting, that of leaves 184 to 189 and 
that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in 
the same handwriting, changes first gradually into that of leaves 184 
to 189 and then (verso) into a third more rapid cursive writing which 
is the same as that of the greater part of the scholia, and also as that 
of leaves 243 and 282, although, as these leaves are of different 
material, the look of the writing and of the ink seems altered. 
There are corrections both by the first and a second hand, and scholia 
by many hands. On the whole, in spite of the apparent diversity of 
handwriting in the MS., it is probable that the whole of it was written 
at about the same time, and it may (allowing for changes of material, 
ink etc) even have been written by the same man. It is at least 
certain that, when the Laurentian ms. xxviii, 6 was copied from it, the 
whole MS, was in the condition in which it is now, except as regards 
the later scholia and leaves 283 to 292 which are not in the Laurentian 
MS., that MS. coming to an end where the Phaenomena breaks off 
abruptly in V, Hence Heiberg attributes the whole MS. to the izthc. 

But it was apparently in two volumes originally, the first con- 
sisting of leaves 1 to 183 ; and it is certain that it was not all copied 
at the same time or from one and the same original. For leaves 
1 84 to 202 were evidently copied from two MSS, different both from 
one another and from that from which the rest was copied. Leaves 
184 to the middle of leaf 189 (recto) must have been copied from a 
MS. similar to P, as is proved by similarity of readings, though not 
from F itself. The rest, up to leaf 202, were copied from the Bologna 
MS. (b) to be mentioned below. It seems clear that the content of 
leaves 1 84 to 202 was supplied from other MSS. because there was a 
lacuna in the original from which the rest of V was copied. 



1 See Pauty-Wissowa, Ktal- Encyclopedic dtr class. AUertitiHSWitunickaft, vol. [I., 1896, 
dberg, vol. v. pp. xxts — xxxiii. * 



**fc 



ch. v] THE TEXT 49 

Heiberg sums up his conclusions thus. The copyist of V first 
copied leaves i to 183 from an original in which two quaterntones 
were missing (covering from the middle of Eucl. X. 105 to near the 
end of XI. 31). Noticing the lacuna he put aside one quatemio of the 
parchment used up to that point. Then he copied onwards from 
the end of the lacuna in the original to the end of the Pkaenomena. 
After this he looked about him for another ms. from which to fill up 
the lacuna; finding one, he copied from it as far as the middle of leaf 
189 (recto). Then, noticing that the MS. from which he was copying 
was of a different class, he had recourse to yet another MS. from which 
he copied up to leaf 302, At the same time, finding that the lacuna 
was longer than he had reckoned for, he had to use twelve more 
leaves of a different parchment in addition to the quatemio which he 
had put aside. The whole MS. at first formed two volumes (the first 
containing leaves 1 to 1 83 and the second leaves 1 84 to 282) ; then, 
after the last leaf had perished, the two volumes were made into one 
to which two more quaterniones were also added. A few leaves of the 
latter of these two have since perished, 

(5) b = MS. numbered 18 — 19 in the Communal Library at 

Bologna, in two volumes, 4to ; nth c. 
This MS. has scholia in the margin written both by the first hand 
and by two or three later hands ; some are written by the latest hand, 
Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 
14th c.) who owned the MS. at one time. It contains (a) in 14 quater- 
niones the definitions and the enunciations (without proofs) of the 
Elements I. — XIII, and of the Data, {b) in the remainder of the 
volumes the Proem to Geometry (published among the Variae 
Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) 
followed by the Elements I. — XIII. (part of XIII. 18 to the end being 
missing), and then by part of the Data (from the last three words of 
the enunciation of Prop. 38 to the end of the penultimate clause in 
Prop. 87, ed. Menge). From xi. 36 inclusive to the end of xn. this 
MS. appears to represent an entirely different recension. Heiberg is 
compelled to give this portion of b separately in an appendix. He 
conjectures that it is due to a Byzantine mathematician who thought 
Euclid's proofs too long and tiresome and consequently contented 
himself with indicating the course followed 1 . At the same time this 
Byzantine must have had an excellent MS. before him, probably of the 
ante-Theonine variety of which the Vatican Ms. 190 (P) is the sole 
representative. 

(6) p = Paris MS. 2466, 41.0; 12th c 

This manuscript is written in two hands, the finer hand occupying 
leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 
64, which are of the same parchment as the earlier leaves, and leaves 
65 to 239, which are of a thinner and rougher parchment showing 
traces of writing of the 8th — 9th c (a Greek version of the Old 
Testament). The ms, contains the Elements 1. — xm. and some scholia 
after Books XL, XII. and xm. 

1 ZtUs<hriflfur Math, u. Pkysik, xxix., hiit.-litt. Abtheilung, p. [3. 



50 INTRODUCTION [ch, v 

(7) 1 ~ Paris MS. 2344, folio ; 12th c. 

It is written by one hand but includes scholia by many hands. 
On leaves 1 to 16 (recto) are scholia with the same title as that found 
by Wachsmuth in a Vatican MS. and relied upon by him to prove that 
Proclus continued his commentaries beyond Book I. 1 Leaves 17 to 
357 contain the Elements I. — Xltl. (except that there is a lacuna from 
the middle of VIII. 25 to the «&<rn? of ix. 14) ; before Books VII. and 
X. there are some leaves filled with scholia only, and leaves 358 to 366 
contain nothing but scholia. 

(8) Heiberg also used a palimpsest in the British Museum (Add. 
17211). Five pages are of the 7th— 8th c. and are contained (leaves 
49 — 53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 
9th a ; half of leaf 50 has perished. The leaves contain various frag- 
ments from Book x, enumerated by Heiberg, Vol. HI., p. v, and nearly 
the whole of XIII. 14. 

Since his edition of the Elements was published, Heiberg has 
collected further material bearing on the history of the text". Besides 
giving the results of further or new examination of MSS.. he has 
collected the fresh evidence contained in an-Nairizi's commentary, 
and particularly in the quotations from Heron's commentary given in 
it (often word for word), which enable us in several cases to trace 
differences between our text and the text as Heron had it, and to 
identify some interpolations which actually found their way into the 
text from Heron's commentary itself; and lastly he has dealt with 
some valuable fragments of ancient papyri which have recently come 
to light, and which are especially important in that the evidence drawn 
from them necessitates some modification in the views expressed in 
the preface to Vol. V. as to the nature of the changes made in Theon's 
recension, and in the principles laid down for differentiating between 
Theon's recension and the original text, on the basis of a comparison 
between P and the Theonine MSS. alone. 

The fragments of ancient papyri referred to are the following. 

1. Papyrus Herculanensis No. 1061'. 

This fragment quotes Def. 15 of Book I. in Greek, and omits the 
words jj icaXelrai irepupepem, "which is called the circumference," 
found in all our MSS., and the further addition Trpbt t>)v toD kvk\ov 
V€ptif>epeiav also found in practically all the MSS. Thus Heiberg's 
assumption that both expressions are interpolations is now conSrmed 
by this oldest of all sources. 

2. The Oxyrhynchus Papyri \. p. 58, No. xxix. of the 3rd or 4U1 c. 
This fragment contains the enunciation of Eucl. II. 5 (with figure, 

apparently without letters, immediately following, and not, as usual in 
our MSS., at the end of the proof) and before it the part of a word 
irept#xpp£ belonging to II. 4 (with room for -v<p 6p8oyaiviq>' cmep ehu 

1 [dt r]A rod EtixXtloav rrotxtia Tpo\at*fia*6tti*& i* w6r Hp&cXou wopdiijv *al rar' /rt- 
rdp^r. Cf. p. 33, note 8, Above. 

■ Heiberg, Paralipemtna ;u liuklid in Htrma, XXXV MI., 190 J, pp. 46 — 74, ifil — aoi, 
311—356. 

1 Described by Heiberg in Overrigt ever dtt kngi, dansks Vtdtnskabtrittt Sthkabs 
Forhandiingtr, rooo, p. 161. 



em v] THE TEXT 51 

Sei|o( and a stroke to mark the end), showing that the fragment had 
not the Porism which appears in all the Theonine MSS. and (in a later 
hand) in P, and thereby confirming Hetberg's assumption that the 
Porism was due to Theon. 

3, A fragment in Fayum towns and their papyri, p. 96, No. IX. of 
2nd or 3rd c. 

This contains I. 39 and I. 41 following one another and almost 
complete, showing that I. 40 was wanting, whereas it is found in all 
the MSS. and is recognised by Prod us. Moreover the text of the 
beginning of I. 39 is better than ours, since it has no double Biopwfios 
but omits the first (" I say that they are also in the same parallels ") 
and has " and" instead of "for let AD be joined " in the next sentence. 
It is clear that I. 40 was interpolated by someone who thought there 
ought to be a proposition following I. 39 and related to it as I. 38 is 
related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses 1. 40, 
and therefore was not likely to include it The same interpolator 
failed to realise that the words " let AD be joined" were part of the 
eK^ca-is or setting-out, and took them for the *oTocr*einj or " construc- 
tion " which generally follows the 8(op«r^>? or " particular statement " 
of the conclusion to be proved, and consequently thought it necessary 
to insert a &topurfi,6<! before the words. 

The conclusions drawn by Heiberg from a consideration of 
particular readings in this papyrus along with those of our MSS. will 
be referred to below. 

We now come to the principles which Heiberg followed, when 
preparing his edition, in differentiating the original text from the 
Theonine recension by means of a comparison of the readings of P 
and of the Theonine MSS. The rules which he gives are subject to a 
certain number of exceptions (mostly in cases where one Ms. or the 
other shows readings due to copyists' errors), but in general they may 
be relied upon to give conclusive results. 

The possible alternatives which the comparison of P with the 
Theonine mss. may give in particular passages are as follows : 

I. There may be agreement in three different degrees. 

(1) P and all the Theonine MSS. may agree. 

In this case the reading common to all, even if it is corrupt or 
interpolated, is more ancient than Theon, i.e. than the 4th c. 

(2) P may agree with some (only) of the Theonine MSS. 

In this case Heiberg considered that the latter give the true 
reading of Theon's recension, and the other Theonine MSS. have 
departed from it 

(3) P and one only of the Theonine MSS may agree. 

In this case too Heiberg assumed that the one Theonine MS. which 
agrees with P gives the true Theonine reading, and that this rule even 
supplies a sort of measure of the quality and faithfulness of the 
Theonine MSS. Now none of them agrees alone \.ith P in preserving 
the true reading so often as F. Hence F must be held to havt pre- 
served Theon's recension more faithfully than the other Theonine MSS. ; 
and it would follow that in those portions where F fails us P must 



5» INTRODUCTION [en. v 

carry rather more weight even though it may differ from the Theortine 
MSS. BVpq. (Heiberg gives many examples in proof of this, as of his 
main rules generally, for which reference must be made to his Prole- 
gomena in Vol. V.) The specially close relation of F and P is also 
illustrated by passages in which they have the same errors; the 
explanation of these common errors (where not due to accident) is 
found by Heiberg in the supposition that they existed, but were not 
noticed by Theon, in the original copy in which he made his changes. 

Although however F is by far the best of the Theonine MSS,, there 
are a considerable number of passages where one of the others (B, V, 
p or q) alone with P gives the genuine reading of Theon 's recension. 

As the result of the discovery of the papyrus fragment containing 
l. 39, 41, the principles above enunciated under (2) and (3) are found 
by Heiberg to require some qualification. For there is in some cases 
a remarkable agreement between the papyrus and the Theonine MSS. 
(some or all) as against P. This shows that Theon took more trouble 
to follow older MSS., and made fewer arbitrary changes of his own, 
than has hitherto been supposed. Next, when the papyrus agrees 
with some of the Theonine MSS. against P, it must now be held that 
these mss, (and not, as formerly supposed, those which agree with P) 
give the true reading of Theon. If it were otherwise, the agreement 
between the papyrus and the Theonine MSS. would be accidental: but 
it happens too often for this. It is clear also that there must have 
been contamination between the two recensions ; otherwise, whence 
could the Theonine mss. which agree with P and not with the papyrus 
have got their readings? The influence of the P class on the Theonine 
F is especially marked. 

II. There may be disagreement between P and all the Theonine 
MSS. 

The following possibilities arise, 

(1) The Theonine MSS. differ also among themselves. 

In this case Heiberg considered that P nearly always has the true 
reading, and the Theonine MSS. have suffered interpolation in different 
ways after Theon's time. 

(2) The Theonine MSS. all combine against P. 

In this case the explanation was assumed by Heiberg to be one or 
other of the following. 

(<*) The common reading is due to an error which cannot be 
imputed to Theon (though it may have escaped him when putting 
together the archetype of his edition); such error may either have 
arisen accidentally in all alike, or (more frequently) may be 
referred to a common archetype of all the MSS. 

(fi) There may be an accidental error in P ; e.g. something 
has dropped out of P in a good many places, generally through 

6fiOlOT&€VTOV 

(<y) There may be words interpolated in P. 
(0) Lastly, we may have in the Theonine MSS. a change made 
by Theon himself. 
(The discovery of the ancient papyrus showing readings agreeing 



ch. v] THE TEXT 5S 

with some, or with all, of the Theonine MSS. against P now makes tt 
necessary to be very cautious in applying these criteria.) 

It is of course the last class (8) of changes which we have to 
investigate in order to get a proper idea of Theon's recension. 

Heiberg first observes, as regards these, that we shall find that 
Theon, in editing the Elements, altered hardly anything without some 
reason, often inadequate according to our ideas, but still some reason 
which seemed to htm sufficient Hence, in cases of very slight differ- 
ences where both the Theonine MSS. and P have readings good and 
probable in themselves, Heiberg is not prepared to put the differences 
down to Theon. In those passages where we cannot see the least 
reason why Theon, if he had the reading of P before him, should have 
altered it, Heiberg would not at once assume the superiority of P 
unless there was such a consistency in the differences as wouW indicate 
that they were due not to accident but to design. In the absence of 
such indications, he thinks that the ordinary principles of criticism 
should be followed and that proper weight should be attached to the 
antiquity of the sources. And it cannot be denied that the sources of 
the Theonine version are the more ancient. For not only is the 
British Museum palimpsest (L), which is intimately connected with 
the rest of our MSS., at least two centuries older than P, but the other 
Theonine M5S. are so nearly allied that they must be held to have 
had a common archetype intermediate between them and the actual 
edition of Theon ; and, since they themselves are as old as, or older 
than P, their archetype must have been much older. Heiberg gives 

Spp. xlvi, xlvii) a list of passages where, for this reason, he has 
bl lowed the Theonine MSS. in preference to P, 

It has been mentioned above that the copyist of P or rather of its 
archetype wished to give an ancient recension. Therefore (apart from 
clerical errors and interpolations) the first hand in P may be relied 
upon as giving a genuine reading even where a correction by the first 
hand has been made at the same time. But in many places the first 
hand has made corrections afterwards ; on these occasions he must 
have used new sources, e.g. when inserting the scholia to the first 
Book which P alone has, and in a number of passages he has made 
additions from Theonine MSS. 

We cannot make out any " family tree " for the different Theonine 
MSS. Although they all proceeded from a common archetype later 
than the edition of Theon itself, they cannot have been copied one 
from the other ; for, if they har' been, how could it have come about 
that in one place or other each of them agrees alette with P in pre- 
serving the genuine reading .' Moreover the great variety in their 
agreements and disagreements indicates that they have all diverged 
to about the same extent from their archetype. As we have seen that 
P contains corrections from the Theonine family, so they show correc- 
tions from P or other MSS. of the same family. Thus V has part of 
the lacuna in the ms. from which it was copied filled up from a MS. 
similar to P, and has corrections apparently derived from the same ; 
the copyist, however, in correcting V, also used another MS. to which 



54 INTRODUCTION [ch. v 

he alludes in the additions to IX. 19 and 30 (and also on x. 23 Por.): 
"in the book of the Ephesian (this) is not found," Who this Ephesian 
of the 12th c. was, we do not know. 

We now come to the alterations made by Theon in his edition of 
the Elements. I shall indicate classes into which these alterations 
may be divided but without details (except in cases where they affect 
the mathematical content as distinct from form or language pure and 
simple) 1 . 

I. Alterations made by Theon where he found, or thought he found, 
mistakes in the original. 

1. Real blots in the original which Theon saw and tried to 
remove. 

(a) Euclid has a porism (corollary) to VI. 19, the enunciation 
of which speaks of similar and similarly described figures though the 
proposition itself refers only to triangles, and therefore the porism 
should have come after VI. 20. Theon substitutes triangle for figure 
and proves the more general porism after vi. 2a 

(o) In IX. 19 there is a statement which is obviously incorrect. 
Theon saw this and altered the proof by reducing four alternatives to 
two, with the result that it fails to correspond to the enunciation even 
with Theon 's substitution of "if" for "when" in the enunciation. 

(c) Theon omits a porism to IX. 11, although it is necessary for 
the proof of the succeeding proposition, apparently because, owing to 
an error in the text (kcitcL tov corrected by Heiberg into «ri to), he 
could not get out of it the right sense. 

(d) I should also put into this category a case which Heiberg 
classifies among those in which Theon merely fancied that he found 
mistakes, viz. the porism to V. 7 stating that, if four magnitudes are 
proportional, they are proportional inversely. Theon puts this after 
v, 4 with a proof, which however has no necessary connexion with 
V. 4 but is obvious from the definition of proportion, 

(e) I should also put under this head XI. I, where Euclid's argu- 
ment to prove that two straight lines cannot have a common segment 
is altered. 

2. Passages which seemed to Theon to contain blots, and which 
he therefore set himself to correct, though more careful consideration 
would have shown that Euclid's words are right or at least may be 
excused and offer no difficulty to an intelligent reader. Under this 
head come : 

(a) an alteration in III. 24. 

{b) a perfectly unnecessary alteration, in VI, 14, of " equiangular 
parallelograms " into " parallelograms having one angle equal to one 
angle," where Theon followed the false analogy of VI. 15. 

(c) an omission of words in V. 26, owing to his having been mis- 
led by a wrong figure. 

(d) an alteration of the order of xi. Deff. 27, 28. 

(e) the substitution of " parallelepiped a I solid " for " cube " in XI. 

1 Exhaustive details under all the different heads are given by Heiberg (Vol. v. 
pp. lii — Uxv). 



ch. v] THE TEXT $S 

38, because Theon observed, correctly enough, that it was true of the 
parallelepipedal solid in general as well as of the cube, but failed to 
give weight to the fact that Euclid must have given the particular 
case of the cube for the simple reason that that was all he wanted for 
use in XIII, 17. 

{/) the substitution of the letter 4> for fl ( V for Z in my figure) 
because he saw that the perpendicular from K to B4> would fall on <£ 
itself, so that <t>, tl coincide. But, if the substitution is made, it should 
be proved that 4>, £1 coincide. Euclid can hardly have failed to notice 
the fact, but it may be that he deliberately ignored it as unnecessary 
for his purpose, because he did not .want to lengthen his proposition 
by giving the proof. 

I I. Emendations intended to improve the form or diction of Euclid. 
Some of these emendations of Theon affect passages of appreciable 

length. Heiberg notes about ten such passages ; the longest is 
in Eucl. xif. 4 where a whole page of Heiberg's text is affected and 
Theon's version is put in the Appendix. The kind of alteration may 
be illustrated by that in ix. r 5 where Euclid uses successively the 
propositions VII, 24, 25, quoting the enunciation of the former but not 
of the latter ; Theon does exactly the reverse. In a few of the cases 
here quoted by Heiberg, Theon shortened the original somewhat. 

But, as a rule, the emendations affect only a few words in each 
sentence. Sometimes they are considerable enough to alter the con- 
formation of the sentence, sometimes they are trifling alterations 
"more magistellorum ineptorum" and unworthy of Theon. Generally 
speaking, they were prompted by a desire to change anything which 
was out of the common in expression or in form, in order to reduce 
the language to one and the same standard or norm. Thus Theon 
changed the order of words, substituted one word for another where 
the latter was used in a sense unusual with Euclid (e.g. i-ireihijirep, 
" since," for on. in the sense of " because "), or one expression for 
another in like circumstances (e.g. where, finding "that which was 
enjoined would be done" in a theorem, VII. 31, and deeming the phrase 
more appropriate to a problem, he substituted for it " that which is 
sought would be manifest"; probably also and for similar reasons he 
made certain variations between the two expressions usual at the end 
of propositions otrep eSei Setl-ai and oirep (Bet iroifjerat, quod erat 
demonstrandum and quod erat faciendum). Sometimes his alterations 
show carelessness in the use of technical terms, as when he uses 
aTTTtaSai (to meet) for t4>d-n-T£<r$at, (to touch) although the ancients 
carefully distinguished the two words. The desire of keeping to a 
standard phraseology also led Theon to omit or add words in a 
number of cases, and also, sometimes, to change the lettering of 
figures. 

But Theon seems, in editing the Elements, to have bestowed the 
most attention upon 

III. Additions designed to supplement or explain Euclid. 

First, he did not hesitate to interpolate whole propositions where 
he thought there was room or use for them. We have already 



56 INTRODUCTION [ch. v 

mentioned the addition to VI. 33 of the second part relating to sectors, 
for which Theon himself takes credit in his commentary on Ptolemy. 
Again, he interpolated the proposition commonly known as vn. 22 
(ex aequo in proportions perturbata for numbers, corresponding to V. 23), 
and perhaps also vn. 20, a particular case of VII. 19 as VI. 17 is of VI. 
16. He added a second case to VI. 27, a porism to II. 4, a second 
porism to III. 1 6, and a lemma after X. 12 ; perhaps also the porism 
to V. 19 and the first porism to vi. 2a He also inserted alternative 
proofs here and there, e.g. in II. 4 (where the alternative differs little 
from the original) and in vn. 3 1 ; perhaps also in X. 1 , 6, and 9. 

Secondly, he sometimes repeats an argument where Euclid had 
said " For the same reason," adds specific references to points, 
straight lines etc. in the figures in order to exclude the possibility 
of mistake arising from Euclid's reference to them in general terms, 
or inserts words to make the meaning of Euclid more plain, e.g. 
componendo and alternately, where Euclid had left them out. Some- 
times he thought to increase by his additions the mathematical 
precision of Euclid's language in enunciations or elsewhere, sometimes 
to make smoother and clearer things which Euclid had expressed 
with unusual brevity and harshness or carelessness, in reliance on the 
intelligence of his readers. 

Thirdly, he supplied intermediate steps where Euclid's argument 
seemed too rapid and not easy enough to follow. The form of these 
additions varies ; they are sometimes placed as a definite intermediate 
step with " therefore " or " so that," sometimes they are additions to 
the statement of premisses, sometimes phrases introduced by "since," 
"for" and the like, after the inference. 

Lastly, there is a very large class of additions of a word, or one 
or two words, for the sake of clearness or consistency. Heiberg 
gives a number of examples of the addition of such nouns as 
"triangle," "square," "rectangle," "magnitude," "number," "point," 
"side," "circle," "straight line," "area" and the like, of adjectives 
such as " remaining," " right," " whole," " proportional," and of other 
parts of speech, even down to words like " is " (4(ttC) which is added 
600 times, Si;, upa, ftiv, yap, kcU and the like. 

IV, Omissions by Theon. 

Heiberg remarks that, Theon's object having been, as above 
shown, to amplify and explain Euclid, we should not naturally have 
expected to find him doing much in the contrary process of com- 
pression, and it is only owing to the recurrence of a certain sort of 
omissions so frequently (especially in the first Books) as to exclude 
the hypothesis of their being all due to chance that we are bound to 
credit him, with alterations making for greater brevity. We have 
seen, it is true, that he made omissions as well as additions for the 
purpose of reducing the language to a certain standard form. But 
there are also a good number of cases where in the enunciation of 
propositions, and in the exposition (the re-statement of them with 
reference to the figure), he has left out words because, apparently, 
he regarded Euclid's language as being too careful and precise. 



ch. v] THE TEXT 57 

Again, he is apparently responsible for the frequent omission of the 
words oirtp ekei Setfot (or -iratijo-ai), Q.E.D. (or F.>. at the end of 
propositions. This is often the case at the end of porisms, where, 
in omitting the words, Theon seems to have deliberately departed 
from Euclid's practice. The MS. P seems to show clearly that, where 
Euclid put a porism at the end of a proposition, he omitted the 
Q.E.D. at the end of the proposition but inserted it at the end of the 
porism, as if he regarded the latter as being actually a part of the 
proposition itself. As in the Theonine MSS. the Q.ED, is generally 
omitted, the omission would seem to have been due to Theon. 
Sometimes in these cases the Q.E.D. is interpolated at the end of the 
proposition. 

Heiberg summed up the discussion of Theon 's edition by the 
remark that Theon evidently took no pains to discover and restore 
from MSS. the actual words which Euclid had written, but aimed 
much more at removing difficulties that might be feit by learners 
in studying the book. His edition is therefore not to be compared 
with the editions of the Alexandrine grammarians, but rather with 
the work done by Eutocius in editing 1 Apollonius and with an 
interpolated recension of some of the works of Archimedes by a 
certain Byzantine, Theon occupying a position midway between these 
two editors, being superior to the latter in mathematical knowledge 
but behind Eutocius in industry (these views now require to be some- 
what modified, as above stated). But however little Theon's object 
may be approved by those of us who would rather know the 
ipsissima verba of Euclid, there is no doubt that his work was 
approved by his pupils at Alexandria for whom it was written ; and 
his edition was almost exclusively used by later Greeks, with the 
result that the more ancient text is only preserved to us in one MS. 

As the result of the above investigation, we may feel satisfied 
that, where P and the Theonine MSS. agree, they give us (except in a 
few accidental instances) Euclid as he was read by the Greeks of 
the 4th c. But even at that time the text had been passed from 
hand to hand through more than six centuries, so that it is certain 
that it had already suffered changes, due partly to the fault of 
copyists and partly to the interpolations of mathematicians. Some 
errors of copyists escaped Theon and were corrected in some MSS. 
by later hands. Others appear in all our MSS. and, as they cannot 
have arisen accidentally in all, we must put them down to a common 
source more ancient than Theon. A somewhat serious instance is 
to be found in III. % ; and the use of aTrrka&w for e<j>aTrr4<r0<i> in the 
sense of " touch " may also be mentioned, the proper distinction 
between the words having been ignored as it was by Theon also. 
But there are a number of imperfections in the ante-Theonine text 
which it would be unsafe to put down to the errors of copyists, those 
namely where the good MSS. agree and it is not possible to see any 
motive that a copyist could have had for altering a correct reading. 
In these cases it is possible that the imperfections are due to a 
certain degree of carelessness on the part of Euclid himself; for it 



58 INTRODUCTION [ch. v 

is not possible " Euclidem ab omni naevo vindicare," to use the 
words of Saccheri", and consequently Slmson is not right in attributing 
to Theon and other editors all the things in Euclid to which mathe- 
matical objection can be taken. Thus, when Euclid speaks of " the 
ratio compounded of the sides" for "the ratio compounded of the 
ratios of the sides," there is no reason for doubting that Euclid himself 
is responsible for the more slip-shod expression. Again, in the Books 
XI. — Xin. relating to solid geometry there are blots neither few 
nor altogether unimportant which can only be attributed to Euclid 
himself; and there is the less reason for hesitation in so attributing 
them because solid geometry was then being treated in a thoroughly 
systematic manner for the first time. Sometimes the conclusion 
(ovfvn-ipaa-pa) of a proposition does not correspond exactly to the 
enunciation, often it is cut short with the words teal t« i^rjf "and the 
rest" (especially From Book X. onwards), and very often in Books viil., 
IX. it is omitted. Where all the MSS. agree, there is no ground for 
hesitating to attribute the abbreviation or omission to Euclid; though, 
of course, where one or more mss. have the longer form, it must be 
retained because this is one of the cases where a copyist has a 
temptation to abbreviate. 

Where the true reading is preserved in one of the Theonine MSS. 
alone, Heiberg attributes the wrong reading to a mistake which arose 
before Theon's time, and the right reading of the single MS. to a 
successful correction. 

We now come to the most important question of the Interpolations 
introduced before Theon's time. 

I. Alternative proofs or additional cases. 

It is not in itself probable that Euclid would have given two 
proofs of the same proposition ; and the doubt as to the genuineness 
of the alternatives is increased when we consider the character of 
some of them and the way in which they are introduced. First of 
all, we have those of VI. 20 and XII. 1 7 introduced by " we shall prove 
this otherwise more readily (irpox^pOTepov)" or that of X. 90 " it is 
possible to prove more shortly (trvvTopcoTepov)." Now it is impossible 
to suppose that Euclid would have given one proof as that definitely 
accepted by him and then added another with the express comment 
that the latter has certain advantages over the former. Had he con- 
sidered the two proofs and come to this conclusion, he would have 
inserted the latter in the received text instead of the former. These 
alternative proofs must therefore have been interpolated. The same 
argument applies to alternatives introduced with the words "or even 
thiis " (tj Kal ovtbw), " or even otherwise " (1} teal a\ka><i). Under this 
head come the alternatives for the last portions of in. 7, 8 ; and 
Heiberg also compares the alternatives for parts of HI. 3 1 (that the 
angle in a semicircle is a right angle) and xill. 1 3, and the alternative 
proof of the lemma after X. 32. The alternatives to X. I OS and 106, 

1 Kudidis ab &mm veuve vittdfcafus, Mediolani, 1733. 

* Cf, especially the assumption, without proof or definition, of the criterion for ifutU solid 
angles, and the incomplete proof of xtt. 17. 



ch. v] THE TEXT $9 

again, are condemned by the place in which they occur, namely after 
an alternative proof to x. in;. The above alternatives being all 
admitted to be spurious, suspicion must necessarily attach to the few 
others which are in themselves unobjectionable, Heiberg instances 
the alternative proofs to 111.9,111. 10, VI. 30, vi. 31 and XI. 22, observing 
that it is quite comprehensible that any of these might have occurred 
to a teacher or editor and seemed to him, rightly or wrongly, to be 
better than the corresponding proofs in Euclid. Curiously enough, 
Simson adopted the alternatives to III. 9, 10 in preference to the 
genuine proofs. Since Heiberg's preface was written, his suspicion 
has been amply confirmed as regards tit. 10 by the commentary of 
an-Nairlsl (ed. Curtze) which shows not only that this alternative is 
Heron's, but also that the substantive proposition III. 12 in Euclid 
is also Heron's, having been given by him to supplement HI. 1 1 
which must originally have been enunciated of circles "touching one 
another " simply, i.e. so as to include the case of external as well as 
internal contact, though the proof covered the case of internal contact 
only. "Euclid, in the nth proposition," says Heron, " supposed two 
circles touching one another internally and wrote the proposition on 
this case, proving what it was required to prove in it, But I will 
show how it is to be -proved if the contact be external 1 ." This additional 
proposition of Heron's is by way of adding another case, which brings 
us to that class of interpolation. It was the practice of Euclid and 
the ancients to give only one case (generally the most difficult one) 
and to leave the others to be investigated by the reader for himself. 
One interpolation of a second case (VI. 27) is due, as we have seen, 
to Theon, The two extra cases of XI. 23 were manifestly interpolated 
before Theon 's time, for the preliminary distinction of three cases, 
" (the centre) will either be within the triangle LMN, or on one of 
the sides, or outside. First let it be within," is a spurious addition 
(B and V only). Similarly an unnecessary case is interpolated in 
III. 11. 

II. Lemmas, 

Heiberg has unhesitatingly placed in his Appendix to Vol. HI. 
certain lemmas interpolated either by Theon (on X. 13) or later 
writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). 
But we are here concerned with the lemmas found in all the MSS., 
which however are, for different reasons, necessarily suspected. We 
will deal with the Book x. lemmas last. 

(1) There is an a priori ground of objection to those lemmas 
which come after the propositions to which they relate and prove 
properties used in those propositions ; for, if genuine, they would be a 
sign of faulty arrangement such as would not be likely in a systematic 
work so carefully ordered as the Elements. The lemma to VI. 22 is 
one of this class, and there is the further objection to it that in VI. 28 
Euclid makes an assumption which would equally require a lemma 
though none is found. The lemma after XII. 4 is open to the further 
objections that certain altitudes are used but are not drawn in the 

1 An-NairUl, ed. Curtze, p. ill. 



6o INTRODUCTION {ch. v 

figure (which is not in the manner of Euclid), and that a peculiar 
expression " parallelepi pedal solids described on {avaypafyofiGva airo) 
prisms " betrays a hand other than Euclid's. There is an objection on 
the score of language to the lemma aftc XIII. 2, The lemmas on 
xi. 23, XIII. 13, XIII. 18, besides coming after the propositions to 
which they relate, are not very necessary in themselves and, as regards 
the lemma to XIII. 13, it is to be noticed that the writer of a gloss 
in the proposition could not have had it, and the words "as will 
be proved afterwards " in the text are rightly suspected owing to 
differences between the MS. readings. The lemma to XII. 2 also, to 
which Simson raised objection, comes after the proposition ; but, if it 
is rejected, the words " as was proved before " used in XII, 5 and 1 8, 
and referring to this lemma, must be struck out. 

(2) Reasons of substance are fatal to the lemma before X. 60, 
which is really assumed in X. 44 and therefore should have appeared 
there if anywhere, and to the lemma on X. 20, which tries to prove 
what is already stated in X. Def. 4. 

We now come to the remaining lemmas in Book X., eleven in 
number, which come be/ore the propositions to which they relate and 
remove difficulties in the way of their demonstration. That before 
X. 42 introduces a set of propositions with the words " that the said 
irrational straight lines are uniquely divided... we will prove after 
premising the following lemma," and it is not possible to suppose 
that these words are due to an interpolator ; nor are there any 
objections to the lemmas before X. 14, 17, 22, 33, 54, except perhaps 
that they are rather easy. The lemma before X. 10 and X. 10 itself 
should probably be removed from the Elements ; for X. 10 really uses 
the following proposition X. 1 1, which is moreover numbered 10 by 
the firsthand in P, and the words in x. 10 referring to the lemma "for 
we learnt (how to do this)" betray the interpolator. Heiberg gives 
reason also for rejecting the lemmas before x. 19 and 24 with the 
words " in any of the aforesaid ways " (omitted in the Theonine MSs.) 
in the enunciations of X. 19, 24 and in the exposition of X. 20. Lastly, 
the lemmas before x. 29 may be genuine, though there is an addition 
to the second of them which is spurious. 

Heiberg includes under this heading of interpolated lemmas two 
which purport to be substantive propositions, XI. 38 and XIII. 6. These 
must be rejected as spurious for reasons which will be found in detail 
in my notes on XL 37 and XIII. 6 respectively. The latter proposition 
is only quoted once (in xin. 17) ; probably the words quoting it 
(with ypa/ifty instead of ti/ffeta) are themselves interpolated, and 
Euclid thought the fact stated a sufficiently obvious inference from 
xin. 1. 

III. Porisms (or corollaries). 

Most of the porisms in the text are both genuine and necessary ; 
but some are shown by differences- in the MSS. not to be so, e.g. those 
to 1. 15 (though Prod us has it), in. 31 and vi. 20 (For. 2). Sometimes 
parts of porisms are interpolated. Such are the last few lines in 
the porisms to iv. 5, vi. 8 ; the latter addition is proved later by 



ch.v] THE TEXT 61 

means of VI. 4, 8, so that the writer of these proofs could not have had 
the addition to vi. 8 Por. before him. Lastly, interpolators have added 
a sort of proof to some porisms, as though they were not quite 
obvious enough ; but to add a demonstration is inconsistent with the 
idea of a porism, which, according to Proclus, is a by-product of a 
proposition appearing without our seeking it 

IV. Scholia. 

Several interpolated scholia betray themselves by their wording, 
e.g. those given by Heiberg in the Appendix to Book X. and contain- 
ing the words icaket, ixaXeve ("he calls" or "called"); these scholia were 
apparently written as marginal notes before Theon's time, and, being 
adopted as such by Theon, found their way into the text in P and 
some of the Theon ine MSS. The same thing no doubt accounts for 
the interpolated analyses and syntheses to xm. 1 — 5, as to which see 
my note on xin. 1. 

V. Interpolations in Book X. 

First comes the proposition " Let it be proposed to us to show that 
in square figures the diameter is incommensurable in length with the 
side, ' which, with a scholium after it, ends the tenth Book. The form 
of the enunciation is suspicious enough and the proposition, the proof 
of which is indicated by Aristotle and perhaps was Pythagorean, is 
perfectly unnecessary when X, 9 has preceded. The scholium ends 
with remarks about commensurable and incommensurable solids, 
which are of course out of place before the Books on solids. The 
scholiast on Book x. alludes to this particular scholium as being due 
to " Theon and some others." But it is doubtless much more ancient, 
and may, as Heiberg conjectures have been the beginning of 
Apollonius' more advanced treatise on incom men su rabies. Not only 
is everything in Book X. after x. 1 1 5 interpolated, but Heiberg doubts 
the genuineness even of X. 112 — 115, on the ground that X, m 
rounds off the theory of incommensu rabies as we want it in the Books 
on solid geometry, while X 112 — 115 are not really connected with 
what precedes, nor wanted for the later Books, but seem to form the 
starting-point of a new and more elaborate theory of irrationals, 

VI. Other minor interpolations are found of the same character as 
those above attributed to Theon. First there are two places (xi, 35 
and XI. 26) where, after "similarly we shail prove" and "for the same 
reason," an actual proof is nevertheless given. Clearly the proofs are 
interpolated ; and there are other similar interpolations. There 
are also interpolations of intermediate steps in proofs, unnecessary 
explanations and so on, as to which I need not enter into details. 

Lastly, following Heiberg's order, I come to 

VII. Interpolated definitions, axioms etc. 

Apart from VI. Def. 5 (which may have been interpolated by 
Theon although it is found written in the margin of P by the first 
hand), the definition of a segment of a circle in Book 1. is interpolated, 
as is clear from the fact that it occurs in a more appropriate place in 
Book til. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly 
condemned by Simson — perhaps it was taken from Heron — and 



6* INTRODUCTION [ch. v 

Heiberg would reject vii. Def, 10, as to which see my note on that 
definition. Lastly the double definition of a solid angle (XL Def. n) 
constitutes a difficulty. The use of the word iirifydveia suggests that 
the first definition may have been older than Euclid, and he may have 
quoted it from older elements, especially as his own definition which 
follows only includes solid angles contained by planes, whereas the 
other includes other sorts (cf. the words ypaftpSv, ypaftpaZ<;) which are 
also distinguished by Heron (Def. 22). If the first definition had come 
last, it could have been rejected without hesitation : but it is not so 
easy to reject the first part up to and including " otherwise " (aXAw?). 
No difficulty need be felt about the definitions of "oblong," "rhombus," 
and "rhomboid," which are not actually used in the Elements; they 
were no doubt taken from earlier elements and given for the sake of 
completeness. 

As regards the axioms or, as they are called in the text, common 
notions (Kotval evvotai), it is to be observed that Proclus says 1 that 
Apollonius tried to prove "the axioms," and he gives Apollonius' 
attempt to prove Axiom i. This shows at all events that Apollonius 
had some of the axioms now appearing in the text. But how could 
Apollonius have taken a controversial line against Euclid on the 
subject of axioms if these axioms had not been Euclid's to his know- 
ledge? And, if they had been interpolated between Euclid's time 
and his own, how could Apollonius, living so comparatively short a 
time after Euclid, have been ignorant of the fact ? Therefore some of 
the axioms are Euclid's (whether he called them common notions, or 
axioms, as is perhaps more likely since Proclus calls them axioms) : 
and we need not hesitate to accept as genuine the first three discussed 
by Proclus, viz. (i) things equal to the same equal to one another, 
(2) if equals be added to equals, wholes equal, (3) if equals be 
subtracted from equals, remainders equal. The other two mentioned 
by Proclus (whole greater than part, and congruent figures equal) are 
more doubtful, since they are omitted by Heron, Martianus Capella, 
and others. The axiom that "two lines cannot enclose a space" is 
however clearly an interpolation due to the fact that I. 4 appeared to 
require it. The others about equals added to unequals, doubles of 
the same thing, and halves of the same thing are also interpolated ; 
they are connected with other interpolations, and Proclus clearly 
used some source which did not contain them. 

Euclid evidently limited his formal axioms to those, which seemed 
to him most essential and of the widest application ; for he not un- 
frequently assumes other things as axiomatic, e.g. in vii. 28 that, tf a 
number measures two numbers, it measures their difference. 

The differences of reading appearing in Proclus suggest the 
question of the comparative purity of the sources used by Proclus, 
Heron and others, and of our text. The omission of the definition of 
a segment in Book I. and of the old gloss "which is called the cir- 
cumference" in I. Def. 15 (also omitted by Heron, Taurus, Sextus 

1 Proclus, pp. 194, losqq. 



ch. v] THE TEXT 63 

Empiricus and others) indicates that Prod us had better sources than 
we have ; and Heiberg gives other cases where Proclus omits words 
which are in all our MSS. and where Proclus' reading should perhaps 
be preferred. But, except in these instances (where Proclus may have 
drawn from some ancient source such as one of the older com- 
mentaries), Proclus' MS. does not seem to have been among the best. 
Often it agrees with our worst mss., sometimes it agrees with F where 
F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) 
the common reading of Proclus and F must be rejected, thrice only 
does it agree with P alone, sometimes it agrees with P and some 
Theonine mss., and once it agrees with the Theonine mss. against P 
and other sources. 

Of the other external sources, those which are older than Theon 
generally agree with our best MSS., e.g. Heron, allowing for the 
difference in the plan of his definitions and the somewhat free adap- 
tation to his purpose of the Euclidean definitions in Books X., XI. 

Heiberg concludes that the Elements were most spoiled by inter- 
polations about the 3rd c, for Sextus Empiricus had a correct text, 
while Iamblichus had an interpolated one; but doubtless the purer 
text continued for a long time in circulation, as we conclude from the 
fact that our MSS. are free from interpolations already found in 
Iamblichus' MS. 



CHAPTER VI. 



THE SCHOLIA. 

Heiberg has collected scholia, to the number of about 1500, in 
Vol. V. of his edition of Euclid, and has also discussed and classified 
them in a separate short treatise, in which he added a few others 1 . 

These scholia cannot be regarded as doing much to facilitate the 
reading of the Elements. As a rule, they contain only such observa- 
tions as any intelligent reader could make for himself. Among the 
few exceptions are XI. Nos. 33, 35 (where XI. 22, 23 are extended to 
solid angles formed by any number of plane angles), xil. No. 85 
(where an assumption tacitly made by Euclid in XII. 17 is proved), 
IX. Nos. 28, 29 (where the scholiast has pointed out the error in the 
text of IX. 19). 

Nor are they very rich in historical information ; they cannot be 
compared in this respect with Proclus' commentary on Book I. or 
with those of Eutocius on Archimedes and Apollonius. But even 
under this head they contain some things of interest, e.g. II. No. 1 1 
explaining that the gnomon was invented by geometers for the sake of 
brevity, and that its name was suggested by an incidental characteristic, 
namely that "from it the whole is known (yvapl^Tai), either of the 
whole area or of the remainder, when it (the yva>ftmv) is either placed 
round or taken away"; 11. No, 13, also on the gnomon; IV. No. 2 
stating that Book IV. was the discovery of the Pythagoreans ; 
V. No. 1 attributing the content of Book v. to Eudoxus ; x. No. 1 with 
its allusion to the discovery of incommensurability by the Pytha- 
goreans and to Apollonius' work on irrationals; x. No. 62 definitely 
attributing X. 9 to Theaetetus; XIII. No. I about the "Platonic" figures, 
which attributes the cube, the pyramid, and the dodecahedron to the 
Pythagoreans, and the octahedron and icosahedron to Theaetetus. 

Sometimes the scholia are useful in connexion with the settlement 
of the text, (1) directly, e.g. III. No. 16 on the interpolation of the 
word "within" (eWot) in the enunciation of 111, 6, and x. No. 1 
alluding to the discussion by "Theon and some others" of irrational 
"surfaces" and "solids," as well as "lines," from which we may 

1 Heiberg, Om Schelitrne til Eutlids Elemenitr, Kjebenhavn, 1888. The tract is 
written in Danish, but, fortunately for those who do not read Danish easily, the author has 
appended (pp. 70 — 78) a resume in French. 



ch. vi] THE SCHOLIA 65 

conclude that the scholium at the end of Book x, is not genuine ; 

(2) indirectly in that they sometimes throw light on the connexion 
of certain MSS. 

Lastly, they have their historical importance as enabling us to 
judge of the state of mathematical science at the times when they 
were written. 

Before passing to the classification of the scholia, Heiberg remarks 
that we must separate from them a number of additions in the nature 
of scholia which are found in the text of our MSS. but which can, in 
one way or another, be proved to be spurious. As they are found 
both in P and in the Theonine MSS., they must have been in the MSS. 
anterior to Theon (4th c). But they are, in great part, only found in 
the margin of P and the Theonine MSS. ; in V they are half in the 
text and half in the margin. This can hardly be explained except 
on the supposition that these additions were originally (in the MSS. 
before Theory) in the margin, and that Theon kept them there in his 
edition, but that they afterwards found their way gradually into the 
text of P as well as of the Theonine MSS., or were omitted altogether, 
while particular MSS. have in certain places preserved the old arrange- 
ment Of such spurious additions Heiberg enumerates the following: 
the axiom about equals subtracted from unequals, the last lines of the 
porism to vi. 8, second porisms to v. 19 and to vi. 20, the porism 
to hi. 31, vi. Def. 5, various additions in Book X., the analyses and 
syntheses of XI 1 1. 1 — Si an ^ the proposition XIII. 6. 

The two first classes of scholia distinguished by Heiberg are 
denoted by the convenient abbreviations "Schol. Vat." and "Schol. 
Vind." 

I. Schol, Vat. 

It is first necessary^ to set out the letters by which Heiberg 
denotes certain collections of scholia. 

P = Scholia in P written by the first hand. 

B = Scholia in B by a hand of the same date as the MS. itself, 
generally that of Aretha s. 

F = Scholia in F by the first hand. 

Vat = Scholia of the Vatican MS. 204 of the 10th c, which has 
these scholia on leaves 198 — 205 (the end is missing) as an independent 
collection. It does not contain the text of the Elements. 

V c = Scholia found on leaves 283 — 292 of V and written in the 
same hand as that part of the MS. itself which begins at leaf 233. 

Vat 192 = a Vatican MS. of the 14th c. which contains, after 
(l) the Elements I. — XIII. (without scholia), (2) the Data with scholia, 

(3) Marinus on the Data, the Schol. Vat as an independent collection 
and in their entirety, beginning with 1. No. 88 and ending with xm. 
No. 44. 

The Schol. Vat., the most ancient and important collection of 
scholia, comprise those which are found in PBF Vat. and, from VII, 12 
to IX. 15, in PB Vat. only, since in that portion of the Elements 
F was restored by a later hand without scholia; they also include 1, 



66 INTRODUCTION [ch. vi 

No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 
which may be left out because F, here has a different text In F 
and Vat. the collection ends with Book x. ; but it must also include 
Schol. FB of Books xi. — xill., since these are found along with Schol. 
Vat. to Books I. — X. in several MSS. (of which Vat. 192 is one) as a 
separate collection. The Schol. Vat. to Books X. — XIII. are also 
found in the collection V c (where, curiously enough, xill. Nos, 43, 44 
are at the beginning). The Schol. Vat. accordingly include Schol. 
PBV C Vat, 192, and doubtless also those which are found in two of 
these sources. The total number of scholia classified by Heiberg as 
Schol. Vat. is 138. 

As regards the contents of Schol. Vat. Heiberg has the following 
observations. The thirteen scholia to Book I. are extracts made 
from Proclus by a writer thoroughly conversant with the subject, 
and cleverly recast (with some additions). Their author does not 
seem to have had the two lacunae which our text of Proclus has 
(at the end of the note on 1. 36 and the beginning of the next note, 
and at the beginning of the note on I. 43), for the scholia I. Nos. 125 
and 137 seem to fill the gaps appropriately, at least in part. In 
some passages he had better readings than our MSS. have. The rest 
of Schol. Vat. (on Books II. — xill.) are essentially of the same 
character as those on Book 1., containing prolegomena, remarks on 
the object of the propositions, critical remarks on the text, converses, 
lemmas ; they are, in general, exact and true to tradition. The 
reason of the resemblance between them and Proclus appears to be 
due to the fact that they have their origin in the commentary of 
Pappus, of which we know that Proclus also made use. In support 
of the view that Pappus is the source, Heiberg places some of the 
Schol, Vat. to Book X. side by side with passages from the com- 
mentary of Pappus in the Arabic translation discovered by Woepcke 1 ; 
he also refers to the striking confirmation afforded by the fact that 
XII, No. 2 contains the solution of the problem of inscribing in a 
given circle a polygon similar to a polygon inscribed in another circle, 
which problem Eutocius says' that Pappus gave in his commentary 
on the Elements. 

But, on the other hand, Schol. Vat. contain some things which 
cannot have come from Pappus, e.g. the allusion in X. No. 1 to Theon 
and irrational surfaces and solids, Theon being later than Pappus ; 
in. No. 10 about porisms is more like Proclus' treatment of the 
subject than Pappus', though one expression recalls that of Pappus 
about forming (c^Tj/MtTifeo-ftit) the enunciations of porisms like those 
of either theorems or problems. 

The Schol, Vat. give us important indications as regards the 
text of the EUmtntt as Pappus had it. In particular, they show that 
he could not have had in his text certain of the lemmas in Book X. 
For example, three of these are identical with what we find in Schol, 

1 Om Scholitnu til Euklids EUmtnltr, pp. 11, it: cf. Euilid-StHtiien, pp. 170, 171; 
Woepcke, MJmoira prtstnt. & ?Ac«d. dts Seitnnt, 18 56, XIV. p. 6j8sqq. 
* Archimedes, ed. Heiberg, in. p. a8, 19—15. 



CM. VI] THE SCHOLIA 67 

Vat (the lemma to X. 17 = Schol. X. No. 106, and the lemmas to 
X. 54, 60 come in Schol. X. No. 328) ; and it is not possible to suppose. 
that these lemmas, if they were already in the text, would also be 
given as scholia. Of these three lemmas, that before X. 60 has 
already been condemned for other reasons ; the other two, un- 
objectionable in themselves, must be rejected on the ground now 
stated. There were four others against which Heiberg found nothing 
to urge when writing his prolegomena to Vol. v., viz. the lemmas 
before X. 42, X. 14, X. 2Z and X. 33. Of these, the lemma to X. 22 
is not reconcilable with Schol, x. No. 161, which takes up the 
assumption in the text of Eucl. X. 22 as if no lemma had gone before. 
The lemma to X. 42, which, on account of the words introducing it 
(see p. 60 above), Heiberg at first hesitated to regard as an inter- 
polation, is identical with Schol. X. No. 27a It is true that in 
Schol. x. No. 269 we find the words "this lemma has been proved 
before (£p tow ipTrpoaStv), but it shall also be proved now for 
convenience' sake (rov eroinov evexa,)" and it is possible to suppose 
that " before " may mean in Euclid's text before x. 42 ; but a proof 
in that place would surely have been as " convenient " as could be 
desired, and it is therefore more probable that the proof had been 
given by Pappus in some earlier place. (It may be added that the 
lemma to X. 14, which is identical with the lemma to XI. 23, con- 
demned on other grounds, is for that reason open to suspicion.) 

Heiberg's conclusion is that all the lemmas are spurious, and that 
most or alt of them have found their way into the text from Pappus' 
commentary, though at a time anterior to Theon's edition, since 
they are found in all our MSS. This enables us to fix a date for these 
interpolations, namely the first half of the 4th c. 

Of course Pappus had not in his text the interpolations which, 
from the fact of their appearing only in some of our MSS., are seen to 
be later than those above-mentioned. Such are the lemmas which 
are found in the text of V only after X. 29 and X. 31 respectively and 
are given in Heiberg's Appendix to Book X. (numbered 10 and 11). 
On the other hand it appears from Woepcke's tract 1 that Pappus 
already had x, 115 in his text : though it does not follow from this 
that the proposition is genuine but only that interpolations began 
very early. 

Theon interpolated a proposition (or lemma) between X. 12 and 
X. 13 (No, S in Heiberg's Appendix). Schol. Vat. has the same 
thing (X. No. 125). The writer of the scholia therefore did not find 
this lemma in the text. Schol. Vat IX. Nos. 28, 29 show that neither 
did he find in his text the alterations which Theon made in Eucl. IX. 
19; the scholia in fact only agree with the text of P, not with Theon's. 
This suggests that Schol. Vat. were written for use with a MS. of the 
ante-Theonine recension such as F is. This probability is further 
confirmed by a certain independence which P shows in several places 
when compared with the Theon ine MSS. Not only has P better 
readings in some passages, but more substantial divergences; and, 

1 Woepcke, op, rft. p. 702. 



68 INTRODUCTION [eft vi 

in particular, the absence in P of three notes of a historical character 
which are added, wholly or partly from Prod us, in the Theonine MSS. 
attests an independent and more primitive point of view in P. 

In view of the distinctive character of P, it is possible that some 
of the scholia found in it in the first hand, but not in the other 
sources of Schol. Vat., also belong to that collection ; and several 
circumstances confirm this. Schol. XIII. No. 45, found in P only, 
which relates to a passage in Eucl. XIII. 13, shows that certain words 
in the text, though older than Theon, are interpolated ; and, as the 
scholium is itself older than Theon, is headed "third lemma," and 
follows a "second lemma" relating to a passage in the text im- 
mediately preceding, which "second lemma" belongs to Schol. Vat. 
and is taken from Pappus, the "third" in all probability came from 
Pappus also. The same is true of Schol. XII. No. 72 and xm. No. 69, 
which are respectively identical with the propositions vulgo XI. 38 
(Heiberg, A pp. to Book xi., No, 3) and XII I. 6; for both of these 
interpolations are older than Theon. Moreover most of the scholia 
which P in the first hand alone has are of the same character as 
Schol, Vat Thus VII. No. 7 and XIII. No. I introducing Books VII. 
and xm. respectively are of the same historical character as several 
of Schol Vat ; that vil. No. 7 appears in the text of P at the 
beginning of Book VII. constitutes no difficulty. There are a number 
of converses, remarks on the relation of propositions to one another, 
explanations such as XII. No. 89 in which it is remarked that <f>, fl 
in Euclid's figure to xil. 17 {Z, V in my figure) are really the same 
point but that this makes no difference in the proof. Two other 
Schol. P on XII. 17 are connected by their headings with XII. No. 72 
mentioned above, xi. No. 10 (P) is only another form of xi. 
No. 1 1 (B) ; and B often, alone with P, has preserved Schol. Vat 
On the whole Heiberg considers some 40 scholia found in P alone to 
belong to Schol. Vat. 

The history of Schol. Vat. appears to have been, in its main 
outlines, the following. They were put together after 500 A.D., since 
they contain extracts from Proclus, to which we ought not to assign 
a date too near to that of Proclus' work itself; and they must at least 
be earlier than the latter half of the 9th c, in which B was written. 
As there must evidently have been several intermediate links between 
the archetype and B, we must assign them rather to the first half of 
the period between the two dates, and it is not improbable that they 
were a new product of the great development of mathematical studies 
at the end of the 6th c. (Isidorus of Miletus). The author extracted 
what he found of interest in the commentary of Proclus on Book I. 
and in that of Pappus on the rest of the work, and put these extracts 
in the margin of a MS. of the class of P. As there are no scholia to 
I. 1 — 22, the first leaves of the archetype or of one of the earliest 
copies must have been lost at an early date, and it was from that 
mutilated copy that partly P and partly a MS. of the Theonine class 
were taken, the scholia being put in the margin in both. Then the 
collection spread through the Theonine MSS., gradually losing some 



ch. vi] THE SCHOLIA 69 

scholia which could not be read or understood, or which were 
accidentally or deliberately omitted. Next it was extracted from 
one of these MSS. and made into a separate work which has been 
preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided 
into sections, so that ihe scholia to Books X. — xni. were detached 
(V c ). It had the same fate in the mss, which kept the original 
arrangement (in the margin), and in consequence there are some MSS. 
where the scholia to the stereometric Books are missing, those Books 
having come to be less read in the period of decadence. It is from 
one of these MSS. that the collection was extracted as a separate work 
such as we find it in Vat. ( roth c). 

II. The second great division of the scholia is Schol. Vind. 

This title is taken from the Viennese MS. (V), and the letters used 
by Heiberg to indicate the sources here in question are as follows. 

V* = scholia in V written by the same hand that copied the MS. 
itself from fol. 235 onward. 

q = scholia of the Paris MS. 2344 (q) written by the first hand. 

1 = scholia of the Florence ms. Laurent xxvin, 2 written in the 
13th — 14th c, mostly in the first hand, but partly in two later 
hands. 

V b = scholia in V written by the same hand as the first part 
(leaves 1 — 1S3) of the MS. itself; V" wrote his scholia after V". 

q 1 = scholia of the Paris MS. (q) found here and there in another 
hand of early date. 

Schol. Vind. include scholia found in V m q. 1 is nearly related to 
q ; and in fact the three Mss. which, so far as Euclid's text is con- 
cerned, show no direct interdependence, are. as regards their scholia, 
derived from one original. Heiberg proves this by reference to the 
readings of the three in two passages (found in Schol. I, No. 109 and 
X. No, 39 respectively). The common source must have contained, 
besides the scholia found in the three MSS. V a ql, those also which 
are contained in two of them, for it is more unlikely that two of the 
three should contain common interpolations than that a particular 
scholium should drop out of one of them. Besides V" and q, the 
scholia V b and q 1 must equally be referred to Schol. Vind., since the 
greater part of their scholia are found in 1. There is a lacuna in q 
from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements 
Schol. Vind. are represented by VI only, Heiberg gives about 450 
numbers in all as belonging to this collection. 

Schol. Vind. did not all come from one source; this is shown by 
differences of substance, e.g. between X. Nos. 36 and 39, and by 
differences of time of writing : e.g. vi. No. 52 refers at the beginning 
to No. 55 with the words "as the scholium has it" and is therefore 
later than that scholium ; X. No. 247 is also later than x. No. 246. 

The scholia to Book I. are here also extracts from Proclus, but 
more copious and more verbatim than in Schol. Vat. The author 
has not always understood Proclus; and he had a text as bad as 
that of our MSS., with the same lacunae. The scholia to the other 



7 o INTRODUCTION [ch. VI 

Books are partly drawn (i) from Schol. Vat., the MSS. representing 
Schol. Vind, and Schol. Vat. in these cases showing nearly all possible 
combinations; but there is no certain trace in Schol. Vind. of the 
scholia peculiar to P. The author used a copy of Schol. Vat. in the 
form in which they were attached to the Theonine text ; thus Schol. 
Vind. correspond to BF Vat., where these diverge from P, and 
especially closely to B. Besides Schol. Vat., the editors of Schol. 
Vind. used {2) other old collections 0/ scholia of which we find traces 
in B and F; Schol. Vind. have also some scholia common with b. 
The scholia which Schoi. Vind. have in common with BF come from 
two different sources, and were apparently afterwards introduced 
into the other MSS. ; one result of this is that several scholia are 
reproduced twice. 

But, besides the scholia derived from these sources, Schol. Vind. 
contain a large number of others of late date, characterised by in- 
correct language or by triviality of content (there are many examples 
in numbers, citations of propositions used, absurd diroplai, and the 
like). Unlike Schol. Vat, these scholia often quote words from Euclid 
as a heading (in one case a heading is inserted in Schol. Vind. where 
a scholium without the heading is quoted from Schol. Vat, see V. 
No. 14). The explanations given often presuppose very little know- 
ledge on the part of the reader and frequently contain obscurities 
and gross errors. 

Schol. Vind. were collected for use with a MS. of the Theonine 
class; this follows from the fact that they contain a note on the 
proposition vulgo VII. 22 interpolated by Theon (given in Heiberg's 
App. to Vol. II. p. 430), Since the scholium to vn. 39 given in V and 
p in the text after the title of Book VIII. quotes the proposition as 
VII. 39, it follows that this scholium must have been written before 
the interpolation of the two propositions vulgo VII. 20, 22 ; Schol. 
Vind. contain (vn. No. 80) the first sentence of it, but without the 
heading referring to VII. 39. Schol. VII. No. 97 quotes VII. 33 as 
VII. 34, so that the proposition vulgo vn. 22 may have stood in the 
scholiast's text but not the later interpolation vulgo vn. 20 (later 
because only found in B in the margin by the first hand). Of course 
the scholiast had also the interpolations earlier than Theon. 

For the date of the collection we have a lower limit in the date 
(12th c.) of MSS. in which the scholia appear. That it was not much 
earlier than the 12th c. is indicated (1) by the poverty of its contents, 
(2) by the quality of the ms. of Proclus which was used in the 
compilation of it (the Munich MS. used by Friedlein with which the 
scholiast's excerpts are essentially in agreement belongs to the I ith — 
12th c), (3) by the fact that Schol. Vind. appear only in MSS. of the 
12th c. and no trace of them is found in our MSS. belonging to 
the 9th — 10th c. in which Schol. Vat. are found. The collection may 
therefore probably be assigned to the 1 ith c. Perhaps it may be in 
part due to Psellus who lived towards the end of that century : for in 
a Florence MS. (Magliabecch. XI, 53 of the 15 th c.) containing a 
mathematical compendium intended for use in the reading of Aristotle 



ch. vi] THE SCHOLIA 71 

the scholia i. Nos. 40 and 49 appear with the name of Psellus 
attached. 

Schoi. Vind. are not found without the admixture of foreign 
elements in any of our three sources. In 1 there are only very few 
such in the first hand. In q there are several new scholia in the first 
hand, for the most part due to the copyist himself. The collection of 
scholia on Book x. in q (Heiberg's q=) is also in the first hand ; it is 
not original, and it may perhaps be due to Psellus (Maglb. has some 
definitions of Book x. with a heading "scholia of... Michael Psellus 
on the definitions of Euclid's 10th Element" and Schol. X. No. 9), 
whose name must have been attached to it in the common source of 
Maglb. and q ; to a great extent it consists of extracts from Schol. 
Vind. taken from the same source as VI. The scholia q 1 (in an 
ancient hand in q), confined to Book II., partly belong to Schol, Vind. 
and partly correspond to b 1 (Bologna MS.), q* and q b are in one hand 
(Theodorus Antiochita), the nearest to the first hand of q ; they are 
doubtless due to an early possessor of the MS. of whom we know 
nothing more. 

V* has, besides Schol. Vind., a number of scholia which also appear 
in other MSS., one in BFb, some others in P, and some in v (Codex 
Vat. IO38, 13th c.) ; these scholia were taken from a source in which 
many abbreviations were used, as they were often misunderstood by V 1 . 
Other scholia in V" which are not found in the older sources — some 
appearing in V* alone— are also not original, as is proved by mistakes 
or corruptions which they contain ; some others may be due to the 
copyist himself. 

V b seldom has scholia common with the other older sources ; for 
the most part they either appear in V b alone or only in the later 
sources as v or F* (later scholia in F), some being original, others not. 
In Book X. V b has three series of numerical examples, ( 1 ) with Greek 
numerals, (2) alternatives added later, also mostly with Greek numerals, 
(3) with Arabic numerals. The last class were probably the work of 
the copyist himself. These examples (cf. p. 74 below) show the facility 
with which the Byzantines made calculations at the date of the MS. 
(12th c). They prove also that the use of the Arabic numerals (in the 
East- Arabian form) was thoroughly established in the 1 2th c. ; they 
were actually known to the Byzantines a century earlier, since they 
appear, in the first hand, in an Escurial MS. of the 1 1 th c. 

Of collections in other hands in V distinguished by Heiberg (see 
preface to Vol. v.), V 1 has very few scholia which are found in other 
sources, the greater part being original ; V ! , V s are the work of the 
copyist himself; V* are so in part only, and contain several scholia 
from Schol. Vat. and other sources. V* and V J are later than 13th 
— 14th c, since they are not found in f (cod. Laurent XXVHl, 6) which 
was copied from V and contains, besides V" V b , the greater part of 
V 1 and vi. No. 20 of V (in the text). 

In P there are, besides P* (a quite late hand, probably one of the 
old Scriptores Grace i at the Vatican), two late hands (P 1 ), one of 
which has some new and independent scholia, while the other has 



7* INTRODUCTION [ch. vi 

added the greater part of Schol. Vind., partly in the margin and 
partly on pieces of leaves stitched on. 

Our sources for Schol. Vat. also contain other elements. In P 
there were introduced a certain number of extracts from Proclus, to 
supplement Schol. Vat. to Book I. ; they are all written with a 
different ink from that used for the oldest part of the MS., and the 
text is inferior. There are additions in the other sources of Schol. 
Vat. (F and B) which point to a common source for FB and which 
are nearly all found in other mss., and, in particular, in Schol. Vind,, 
which also used the same source ; that they are not assignable to 
Schol. Vat. results only from their not being found in Vat. Of other 
additions in F, some are peculiar to F and some common to it and b; 
but they are not original. F s (scholia in a later hand in F) contains 
three original scholia ; the rest come from V. B contains, besides 
scholia common to it and F, b or other sources, several scholia which 
seem to have been put together by Arethas, who wrote at least a part 
of them with his own hand. 

Heiberg has satis6ed himself, by a closer study of b, that the 
scholia which he denotes by b, ji and b 1 are by one hand ; they are 
mostly to be found in other sources as well, though some are original. 
By the same hand (Theodoras Cabasilas, 15th c.) are also the scholia 
denoted by b", B', b* and B ! . These scholia come in great part from 
Schol. Vind., and in making these extracts Theodoras probably used 
one of our sources, 1, mistakes in which often correspond to those of 
Theodoras. To one scholium is attached the name of Demetrius (who 
must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c); 
but it could not have been written by him, since it appears in B antl 
Schol. Vind. Nor are all the scholia which bear the name of 
Theodoras due to Theodoras himself, though some are so. 

As B' (a late hand in B) contains several of the original scholia of 
b*, B* must have used b itself as his source, and, as all the scholia in 
B* are in b, the latter is also the source of the scholia in B 8 which are 
found in other MSS. B and b were therefore, in the 15th c, in the 
hands of the same person ; this explains, too, the fact that b in a late 
hand has some scholia which can only come from B. We arrive then 
at the conclusion that Theodoras Cabasilas, in the 15th c, owned both 
the MSS. B and b, and that he transferred to B scholia which he had 
before written in b, either independently or after other sources, and 
inversely transferred some scholia from B to b. Further, B' are 
earlier than Theodoras Cabasilas, who certainly himself wrote B* as 
well as b' and b 8 . 

An author's name is also attached to the scholia VI. No. 6 and 
X. No. 223, which are attributed to Maximus Planudes (end of 13th c) 
along with scholia on I. 31, x. 14 and X. 18 found in 1 in a quite late 
hand and published on pp. 46, 47 of Heiberg's dissertation. These 
seem to have been taken from lectures of Planudes on the Elements 
by a pupil who used 1 as his copy. 

There are also in 1 two other Byzantine scholia, written by a late 
hand, and bearing the names Ioannes and Pediasimus respectively ; 



ch. vi] THE SCHOLIA 73 

these must in like manner have been written by a pupil after lectures 
of Ioannes Pediasimus (first half of 14th c), and this pupil must also 
have used 1. 

Before these scholia were edited by Heiberg, very few of them had 
been published in the original Greek. The Basel editio princeps has a 
few (v. No. 1, VI. Nos. 3, 4 and some in Book X.) which are taken, 
some from the Paris MS. (Paris. Gr, 2343) used by Grynaeus, others 
probably from the Venice MS. (Marc. 301) also used by him; one 
published by Heiberg, not in his edition of Euclid but in his paper 
on the scholia, may also be from Venet. 301, but appears also in 
Paris. Gr. 2342. The scholia in the Basel edition passed into the 
Oxford edition in the text, and were also given by August in the 
Appendix to his Vol. II. 

Several specimens of the two series of scholia (Vat. and Vind.) 
were published by C. Wachsmuth {Rhein, Mus. xvm, p. 132 sqq.) 
and by Knoche {Untersuchungen iiber die neu aufgefundenen Scholien 
des Proklus, Herford, 1 865). 

The scholia published in Latin were much more numerous. G. 
Valla {De expetendis et fugiendis rebus, 1 501) reproduced apparently 
some 200 of the scholia included in Heiberg's edition. Several of 
these he obtained from two Modena MSS. which at one time were 
in his possession (Mutin. Ill B, 4 and II E, 9, both of the 15th c.) ; 
but he must have used another source as well, containing extracts 
from other series of scholia, notably Schol. Vind. with which he has 
some 87 scholia in common. He has also several that are new. 

Commandinus included in his translation under the title "Scholia 
antiqua " the greater part of the Schol, Vat. which he certainly 
obtained from a MS. of the class of Vat. 192; on the whole he 
adhered closely to the Greek text. Besides these scholia Com- 
mandinus has the scholia and lemmas which he found in the Basel 
editio princeps, and also three other scholia not belonging to Schol. 
Vat., as well as one new scholium (to Xii. 13) not included in 
Heiberg's edition, which are distinguished by different type and were 
doubtless taken from the Greek MS. used by him along with the 
Basel edition. 

In Conrad Dasypodius' Lexicon matkematicum published in 1573 
there is (on fol. 42—44) "Graecum scholion in definitiones Euclidis 
libri quinti elementorum append ids loco propter pagellas vacantes 
annexum." This contains four scholia, and part of two others, 
published in Heiberg's edition, with some variations of readings, and 
with some new matter added (for which see pp. 64 — 6 of Heiberg's 
pamphlet). The source of these scholia is revealed to us by another 
work of Dasypodius, haaci Monachi Scholia in Euclidis elementorum 
geometriae sex prions tibros per C. Dasypodium in latinum sermonem 
trans lata et in lucem edita (1579). This work contains, besides 
excerpts from Proclus on Book I. (in part closely related to Schol. 
Vind.), some 30 scholia included in Heiberg's edition, several new 
scholia, and the above-mentioned scholia to the definitions of Book v. 
published in Greek in 1573. After the scholia follow " Isaaci Monachi 



74 INTRODUCTION [ch. vi 

prolegomena in Euclidis Elementorum geometriae libros" (two 
definitions of geometry) and " Varia miscellanea ad geometriae cogni- 
tionem necessaria ab Isaaco Monacho collecta " (mostly the same as 
pp. 252, 24 — 272, 27 in the Varieu Collectiones included in Hultsch's 
Heron) ; lastly, a note of Dasypodius to the reader says that these 
scholia were taken "ex clarissimi viri Joannis Sambuciantiquocodice 
manu propria Isaaci Monachi scrip to." Isaak Monachus is doubtless 
Isaak Argyrus, 14th c. ; and Dasypodius used a MS. in which, besides 
the passage in Hultsch's Variae CoIUctiotus, there were a number of 
scholia marked in the margin with the name of Isaak (cf. those in b 
under the name of Theodorus Cabasilas). Whether the new scholia 
are original cannot be decided until they are published in Greek ; but 
it is not improbable that they are at all events independent arrange- 
ments of older scholia. All but five of the others, and all but one of 
the Greek scholia to Book V., are taken from Schol. Vat. ; three of the 
excepted ones are from Schol, Vind., and the other three seem to 
come from F (where some words of them are illegible, but can be 
supplied by means of Mut. Ill B, 4, which has chese three scholia and 
generally shows a certain likeness to Isaak's scholia). 

Dasypodius also published in 1564 the arithmetical commentary 
of Barlaam the monk (14th c.) on Eucl. Book 11., which finds a place 
in Appendix IV. to the Scholia in Heiberg's edition. 

Hultsch has some remarks on the origin of the scholia 1 . He 
observes that the scholia to Book I. contain a considerable portion 
of Geminus' commentary on the definitions and are specially valuable 
because they contain extracts from Geminus only, whereas Proclus, 
though drawing mainly upon him, quotes from others as well. On the 
postulates and axioms the scholia give more than is found in Proclus. 
Hultsch conjectures that the scholium on Book V., No. 3, attributing 
the discovery of the theorems to Eudoxus but their arrangement to 
Euclid, represents the tradition going back to Geminus, and that the 
scholium XIII., No. 1, has the same origin. 

A word should be added about the numerical illustrations of 
Euclid's propositions in the scholia to Book x. They contain a large 
number of calculations with sexagesimal fractions'; the fractions go 
as far as fourth-sixtieths (i/6o*). Numbers expressed in these fractions 
are handled with skill and include some results of surprising accuracy* 

1 Art. " Eukleides" in Pauly-Wissowa's KeaJ-Ettcyflopadit. 

1 Hultsch has written upon these in Biblietluea Matktmatisa, v a , 1904, pp. 155 — 133. 

* Thus v'W) is given (allowing for a slight correction by means of the context) as 5 1 i 1 
46" 10'", which gives for V3 the value t 43 Jj" 13"', being the same value as that given by 
Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly J 8 
is given as 1 49' 41" 10'" 10"", which is equivalent to \'i= IV 141 133s- Hultsch gives 
instances of the various operations, addition, subtraction, etc., carried out in these fractions, 
and shows how the extraction of the square root was effected. Cf. T- L. Heath, History ttj 
Creek Mathematics, t. , pp. 50 — 03- 



CHAPTER VII. 

EUCLID IN ARABIA, 

We are told by Hajl Khalfa' that the Caliph al-MansQr (754-775) 
sent a mission to the Byzantine Emperor as the result of which he 
obtained from him a copy of Euclid among other Greek books, and 
again that the Caliph al-Ma'mun (813-833) obtained manuscripts of 
Euclid, among others, from the Byzantines. The version of the 
Elements by al-Hajjaj b. Yusuf b. Matar is, if not the very first, at 
least one of the first books translated from the Greek into Arabic'. 
According to the Fikrist* it was translated by al-Hajjaj twice ; the 
first translation was known as " Haruni" (" for Harun"), the second 
bore the name "Ma'muni" ("for al-Ma'mun") and was the more trust- 
worthy. Six Books of the second of these versions survive in a Leiden 
MS. (Codex Leidensis 399, 1) now in part published by Besthorn 
and Heiberg*. In the preface to this ms. it is stated that, in the reign 
of Harun ar-Rashid (786-809), al-Hajjaj was commanded by Yahya 
b. Khalid b. Barmak to translate the book into Arabic. Then, when 
al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjaj 
saw that he would secure the favour of al-Ma'mun "if he illustrated 
and expounded this book and reduced it to smaller dimensions. He 
accordingly left out the superfluities, filled up the gaps, corrected or 
removed the errors, until he had gone through the book and reduced 
it, when corrected and explained, to smaller dimensions, as in this 
copy, but without altering the substance, for the use of men endowed 
with ability and devoted to learning, the earlier edition being left in 
the hands of readers." 

The Fikrist goes on to say that the work was next translated by 
Ishaq b. Hunain, and that this translation was improved by Thabit b, 
Qurra. This Abu Ya'qub Ishaq b. Hunain b. Ishaq al-Tbadi (d. 910) 
was the son of the most famous of Arabic translators, Hunain b. Ishaq 
al-'lbadi (809-873), a Christian and physician to the Caliph al- 
Mutawakkil (847-861). There seems to be no doubt that Ishaq, who 

1 ijxicon bibliegr. et tncyclop. ed. FliigeL lilt pp. 91 , pa. 

1 Klamroth, Zeitschrift der Deittschen Morgenltindisehcn Gesellsihaft, XXXV. p, 303. 

3 Fihrist (tr. Suter), p. 16. 

* Codex Leidensis 399, 1. Eueiidis Elementa ex interpretaiione at-Hadschdsehadschii eum 
cgmmmfariis al-Narizii* Elauniae, part C. i. 1893, part t. ii. 1897, part U- L [900, pari ][. 
ii. 1905, part III. i. 1910. 



76 INTRODUCTION [ch. vii 

must have known Greek as well as his father, made his translation 
direct from the Greek. The revision must apparently have been the 
subject of an arrangement between Ishaq and Thabit, as the latter 
died in 901 or nine years before Ishaq, Thabit undoubtedly consulted 
Greek MSS. for the purposes of his revision. This is expressly stated 
in a marginal note to a Hebrew version of the Elements, made from 
Ishaq's, attributed to one of two scholars belonging to the same family, 
viz. either to Moses b. Tibbon (about 1 244- 1 274) or to Jakob b. Machir 
(who died soon after 1306) 1 . Moreover Thabit observes, on the pro- 
position which he gives as ix. 31, that he had not found this proposition 
and the one before it in the Greek but only in the Arabic ; from which 
statement Klamroth draws two conclusions, (1) that the Arabs had 
already begun to interest themselves in the authenticity of the text 
and (2) that Thabit did not alter the numbers of the propositions in 
Ishaq's translation'. The Fihrist also says that Yuhanna al-Qass (i.e. 
" the Priest ") had seen in the Greek copy in his possession the pro- 
position in Book I. which Thabit took credit for, and that this was 
confirmed by Nazlf, the physician, to whom Yuhanna had shown it 
This proposition may have been wanting in Ishaq, and Thabit may 
have added it, but without claiming it as his own discovery*. As 
a fact, t. 45 is missing in the translation by al-Hajjaj. 

The original version of Ishaq without the improvements by Thabit 
has probably not survived any more than the first of the two versions 
by al-Hajjaj ; the divergences between the MSS, are apparently due to 
the voluntary or involuntary changes of copyists, the former class 
varying according to the degree of mathematical knowledge possessed 
by the copyists and the extent to which they were influenced by 
considerations of practical utility for teaching purposes*. Two MSS. 
of the Ishaq-Thabit version exist in the Bodleian Library (No. 279 
belonging to the year 1238, and No. 280 written in 1260-1) 11 ; Books 
I. — XIII. are in the Ishaq-Thabit version, the non- Euclidean Books 
XIV., XV. in the translation of Qusta b. Luqa at-Ba'labakki (d. about 
912). The first of these MSS. (No. 279) is thafctX)) used by Klamroth 
for the purpose of his paper on the Arabian Euclid. The other MS. 
used by Klamroth is (K) Kjobenhavn LXXXI, undated but probably 
of the 13th c, containing Books v. — xv., Boiks V. — X. being in the 
Ishaq-Thabit version, Books XI. — XIII. purporting to be in al-Hajjaj's 
translation, and Books XIV, xv. in the version of Qusta b. Luqa. In 
not a few propositions K and O show not the slightest difference, and, 
even where the proofs show considerable differences, they are generally 
such that, by a careful comparison, it is possible to reconstruct the 
common archetype, so that it is fairly clear that we have in these cases, 
not two recensions of one translation, but arbitrarily altered and 

1 Steinschneider, Ziitschrift fur Math. «. Physik, XXXI., hist. -lilt. Abtheilung, pp. 85, 
86, 90. 

J Klamtoth, p. »79, ■ Steinschneider, p, 88. 

* Klamroth, p. 306. 

1 These MSS. are described by Nicoll and Pusey, Catatogus tod. m$s. orient, bibl. Bca- 
hiattot, pt. u. 1835 (pp. 157—161). 



ch. vh] EUCLID IN ARABIA 77 

shortened copies of one and the same recension 1 . The Bodleian MS. 
No. 280 contains a preface, translated by Nicoll, which cannot be by 
Thibit himself because it mentions Avkenna (980-1037) and other 
later authors. The MS. was written at Maraga in the year 1260-1 and 
has in the margin readings and emendations from the edition of 
Nasiraddln at-Tusi (shortly to be mentioned) who was living at Maraga 
at the time, is it possible that at-Tusl himself is the author of the 
preface*? Be this as it may, the preface is interesting because it 
throws light on the liberties which the Arabians allowed themselves 
to take with the text. After the observation that the book (in spite 
of the labours of many editors) is not free from errors, obscurities, 
redundancies, omissions etc., and is without certain definitions neces- 
sary for the proofs, it goes on to say that the man has not yet been 
found who could make it perfect, and next proceeds to explain 
(1) that Avicenna "cut out postulates and many Definitions" and 
attempted to clear up difficult and obscure passages, (2) that Abu'l 
Wafa al-Buzjanl (939-99?) "introduced unnecessary additions and 
left out many things of great importance and entirely necessary," 
inasmuch as he was too long in various places in Book VI. and too 
short in Book X. where he left out entirely the proofs of the apotomae, 
while he made an unsuccessful attempt to emend XII. 14, (3) that Abu 
Ja'far al-Khazin (d, between 961 and 971) arranged the postulates 
excellently but " disturbed the number and order of the propositions, 
reduced several propositions to one " etc. Next the preface describes 
the editor's own claims' and then ends with the sentences, " But we 
have kept to the order of the books and propositions in the work itself 
(i.e. Euclid's) except in the twelfth and thirteenth books. For we have 
dealt in Book xni, with the (solid) bodies and in Book XII. with the 
surfaces by themselves." 

After Thabit the Fihrist mentions Abu 'Uthman ad-Dimashql as 
having translated some Books of the Elements including Book X. (It 
is Abu 'Uthman's translation of Pappus' commentary on Book X. 
which Woepcke discovered at Paris.) The Fikrist adds also that 
" Nazif the physician told me that he had seen the tenth Book of 
Euclid in Greek, that it had 40 propositions more than the version 
in common circulation which had 109 propositions, and that he had 
determined to translate it into Arabic." 

But the third form of the Arabian Euclid actually accessible to us 
is the edition of AbQ Ja'far Muh. b. Muh. b. al-Hasan Nasiraddln 
at-Tusi (whom we shall call at-Tusi for short), born at Tus ( m 
Khurasan) in 1201 (d. 1274). This edition appeared in two forms, a 
larger and a smaller. The larger is said to survive in Florence only 
(Pal. 272 and 313, the latter MS. containing only six Books) ; this was 
published at Rome in 1 594, and, remarkably enough, some copies of 

1 KlamrOth, pp. jort — S. 

* Steinschneider, p. 98. Heiberg has Quoted the whole of this preface in the Ztitschrift 
fur Math. ft, Phytik, XXIX., hist.-litt. Abth. p. 16. 

* This seems to include a rearrangement of the contents of Books xiv., xv. added to the 
EltmftUs. 



78 INTRODUCTION |ch. vu 

this edition are to be found with 12 and some with 13 Books, some 
with a Latin title and some without'. But the book was printed in 
Arabic, so that Kastner remarks that he will say as much about it as 
can be said about a book which one cannot read*. The shorter form, 
which however, in most MSS., is in 15 Books, survives at Berlin, Munich, 
Oxford, British Museum (974, 1334*, 1335), Paris (2465, 2466), India 
Office, and Constantinople ; it was printed at Constantinople in 
1 80 1, and the first six Books at Calcutta in 1824*. 

At-Tusi's work is however not a trattslatiott of Euclid's text, but a 
re-written Euclid based on the older Arabic translations. In this 
respect it seems to be like the Latin version of the Elements by 
Campanus (Campano), which was first published by Erhard Ratdolt 
at Venice in 1482 (the first printed edition of Euclid*). Campanus 
(13th c.) was a mathematician, and it is likely enough that he allowed 
himself the same liberty as at-TGst in reproducing Euclid. What- 
ever may be the relation between Campanus' version and that of 
Athelhard of Bath (about 1 1 20), and whether, as Curtze thinks*, they 
both used one and the same Latin version of 10th — 1 ith c, or whether 
Campanus used Athel hard's version in the same way as at- T fist used 
those of his predecessors 7 , it is certain that both versions came from 
an Arabian source, as is evident from the occurrence of Arabic words 
in them*. Campanus' version is not of much service for the purpose 
of forming a judgment on the relative authenticity of the Greek and 
Arabian tradition ; but it sometimes preserves traces of the purer 
source, as when it omits Theon's addition to vi. 33'. A curious 
circumstance is that, while Campanus' version agrees with at-Tusi's 
in the number of the propositions in all the genuine Euclidean Books 
except V. and IX., it agrees with At hd hard's in having 34 propositions 
in Book V. (as against 25 in other versions), which confirms the view 
that the two are not independent, and also leads, as Klamroth says, 
to this dilemma: either the additions to Book V. are Athelhard's 
own, or he used an Arabian Euclid which is not known to us". 
Heiberg also notes that Campanus' Books XIV., XV. show a certain 
agreement with the preface to the Thabit-Ishiq version, in which the 
author claims to have (1) given a method of inscribing spheres in the 
five regular solids, (2) carried further the solution of the problem how 

1 Suter, Dit Afatkeniatiier und Asironomen dtr Arobtr, p. iji. The Lai in title- a 
Euilidii tltttteniorum geometrkorum libri trtdtcim. Ex traditions dottissimi Nasiridini 
Ttami nunc frimum arabite itnprtssi. Kumae in typograpbia Medicea MDXCIV. Cum 
licentia superiorum. 

1 Kastner, Gesehiehie der Matntmatik, I. p. 367. 

* Suter has a note that this US. is very old, having been copied from the original in the 
author's lifetime. 

* Suter, p. iji. 

■ Described by Kastner, Gathitkte dtr Mathrmatii, 1. pp. 389 — iog, and by Weiss - 
enbom, Die Uberutxungtn da Etttlid durch Campano und Zamberti, Halle a. S., 1881, 
pp. 1 — J. See also infra. Chapter vni, p, 97. 

■ Sonderabdruck des Jahrtsberitklts titer die Fortiekrittt dtr klastitcktn AUtrlnumt- 
wisstnsehaft vem. Okt. 1S79 — 18B3, Berlin, 1884. 

7 Klamroth, p. 171. 

* Curtw, op. tit. p. 10; Heiberg, Exklid-Studitn, p. 178. 

* Heiberg's Euclid, vol. v. p. ci. l0 Klamroth, pp. 173 — 4. 



ch. vn] EUCLID IN ARABIA 79 

to inscribe any one of the solids in any other and (3) noted the cases 
where this could not be done 1 . 

With a view to arriving at what may be called a common measure 
of the Arabian tradition, it is necessary to compare, in the first place, 
the numbers of propositions in the various Books. Haji Khalfa says 
that al-Hajjaj's translation contained 468 propositions, and Thabit's 
478 ; this is stated on the authority of at-TQsI, whose own edition 
contained 468*. The fact that Thabit's version had 478 propositions 
is confirmed by an index in the Bodleian MS. 279 (called O by 
Klamroth). A register at the beginning of the Codex Leiden sis 399, 1 
which gives Ishaq's numbers (although the translation is that of 
al-FIajjaj) apparently makes the total 479 propositions (the number in 
Book XIV. being apparently 11, instead of the 10 of O 1 ). I subjoin a 
table of relative numbers taken from Klamroth, to which I have added 
the corresponding numbers in August's and Heiberg's editions of the 
Greek text 





The Arabian Euclid 


The Greek Euclid 


Books 


Ishiq 


at-TOsI 


Campaniu 


Gregory 


August 


Heiberg 


1 


48 


48 


48 


48 


48 


48 


11 


14 


14 


>4 


14 


14 


14 


1U 


36 


36 


36 


37 


37 


37 


IV 


16 


16 


16 


16 


16 


16 


V 


*S 


n 


34 


*5 


n 


*s 


VI 


33 


32 


32 


33 


33 


33 


VII 


39 


39 


39 


41 


41 


39 


VIII 


*7 


'S 


IS 


37 


27 


37 


IX 


3« 


36 


39 


36 


36 


36 


X 


109 


107 


107 


117 


116 


115 


XI 


41 


4' 


4< 


40 


40 


39 


XII 


■5 


'5 


'5 


iS 


iS 


18 


XIII 


21 


iS 


■ S 


■ S 


iS 


18 




462 


453 


464 


470 


469 


465 


[XIV 


10 


10 


18 


7 




? 


XV 


6 


6 


'3 


10 








478 


468 


495 


"487 




a 



The numbers in the case of Heiberg include all propositions which 
he has printed in the text ; they include therefore xiii. 6 and in. 12 
now to be regarded as spurious, and X. .1 12 — 115 which he brackets 
as doubtful. He does not number the propositions in Books XIV., XV., 
but I conclude that the numbers in P reach at least 9 in xiv., and 9 
in XV. 

1 Heiberg, Zttisthrift fur Math. u. PAjriii, xxix,, hist.-liu. Atrtheilung, p. 11. 
' Klamroth, p. 17+; Steinachneider, Zatschrift fur Math. u. Physth, XXXI., hid Mitt. 
Abth. p. 9*i. 

■ BeMhorn- Heiberg read " 11?" aa the number, Klamroth had read it as ji (p. 173). 



8o INTRODUCTION [en. vii 

The Fihrist confirms the number 109 for Book X., from which 
K lam roth concludes that Ishaq's version was considered as by far the 
most authoritative. 

In the text of O, Book IV. consists of 1 7 propositions and Book 
XIV. of 12, differing in this respect from its own table of contents ; IV. 
15, 16 in O are really two proofs of the same proposition. 

In al-Hajjaj's version Book I. consists of 47 propositions only, I. 45 
being omitted. It has also one proposition fewer in Book III., the 
Heron ic proposition m. 12 being no doubt omitted. 

In speaking of particular propositions, I shall use Heiberg's 
numbering, except where otherwise stated. 

The difference of 10 propositions between Tha bit- Ishaq and 
at-TusT is accounted for thus : 

(1) The three propositions vi. 12 and X, 28, 39 which both Ishaq 
and the Greek text have are omitted in at-Tusi, 

(2) Ishaq divides each of the propositions xm. 1 — 3 into two, 
making six instead of three in at-Tusi and in the Greek. 

{3) Ishaq has four propositions (numbered by him vm. 24, 2$, 

IX. 30, 3 1) which are neither in the Greek Euclid nor in at-Tusi. 
Apart from the above differences al-iiajjaj (so far as we know), 

Ishaq and at-Tusi agree , but their Euclid shows many differences 
from our Greek text. These differences we will classify as follows 1 . 

1. Prepositions. 

The Arabian Euclid omits VII. 20, 22 of Gregory's and August's 
editions (Heiberg, App. to VoL 11. pp. 428-32) ; vm. 16, 17; X. 7, 8, 
13, 16, 24, 112, 113, 114, besides a lemma vulgo X. 13, the proposition 

X. 1 1 7 of Gregory's edition, and the scholium at the end of the Book 
(see for these Heiberg's Appendix to Vol. III. pp. 382, 408 — 416) ; 

XI. 38 in Gregory and August (Heiberg, App. to Vol. I v. p. 354); 

XII. 6, 13, 14 ; (also all but the first third of Book xv.). 

The Arabian Euclid makes III. 11, 12 into one proposition, and 
divides some propositions (X. 31, 32 ; xi, 31, 34; xin. 1 — 3) into two 
each. 

The order is also changed in the Arabic to the following extent. 
v. 12, 13 are interchanged and the order in Books Vi,, vn, IX. — 

XIII. is : 

VL 1—8, 13, II, 12, 9, IO, 14—17, 19, 20, 18, 21, 22, 24, 26, 23, 
25. 27—30, 32, 31, 33. 

VII. I — 20, 22, 21, 23 — 28, 31, 32, 29, 30, 33 — 39. 

IX i — 13, 20, 14—19, 21 — 25, 27, 26, 28 — 36, with two new pro- 
positions coming before prop. 30. 

x. 1—6, 9—12. IS. 14. i7— 2 3. 26—28, 25, 29—30, 31, 32, 33— 
in, us- 

XL 1—30,31.32,34.33.35—39 

xii. 1 — 5, 7, 9, 8, 10, 12, 11, 15, 16 — 18. 
xm. 1—3, 5. 4, 6, 7, 12, 9, io, 8, 11, 13, is, 14. 16—18. 

1 See KJamroth, pp. 175 — 6, 180, 161 — 4, %\± — 15, 516 ; Heibeig, vol. v. pp. xcvi, xcvii. 



CH.vn] EUCLID IN ARABIA Si 

2. Definitions. 

The Arabic omits the following definitions: iv. Deff. 3 — 7, VII. 
Def. 9 (or io), xi. Deff, 5 — 7, 15, 17. 23, 25—28; but it has the 
spurious definitions VI. Deff. 2, 5, and those of proportion and ordered 
proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges 
v. Deff. 1 1, 12 and also vi. Deff. 3, 4. 

The order of the definitions is also different in Book VII. where, 
after Def. II, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, 
and in Book xi. where the order is 1, 2, 3,4, 8, io, 9, 13, 14, 16, 12, 21, 
22, 18, 19, 20, 11, 24. 

3. Lemmas and porisms. 

All are omitted in the Arabic except the porisms to vi. 8, vin, 2, 
X. 3 ; but there are slight additions here and there, not found in the 
Greek, e.g. in vm. 14, 15 (in K). 

4. Alternative proofs. 

These are all omitted in the Arabic, except that in X, 105, 106 they 
are substituted for the genuine proofs ; but one or two alternative 
proofs are peculiar to the Arabic (vi. 32 and vm. 4, 6). 

The analyses and syntheses to XIII. I — 5 are also omitted in the 
Arabic. 

K lam roth is inclined, on a consideration of all these differences, to 
give preference to the Arabian tradition over the Greek (1) "on 
historical grounds," subject to the proviso that no Greek MS. as 
ancient as the 8th c, is found to contradict his conclusions, which are 
based generally (2) on the improbability that the Arabs would have 
omitted so much if they had found it in their Greek MSS., it being clear 
from the Fihrist that the Arabs had already shown an anxiety for a 
pure text, and that the old translators were subjected in this matter to 
the check of public criticism. Against the " historical grounds," Heiberg 
is able to bring a considerable amount of evidence 1 . First of all there 
is the British Museum palimpsest (L) of the 7th or the beginning of 
the 8th c. This has fragments of propositions in Book X. which are 
omitted in the Arabic ; the numbering of one proposition, which agrees 
with the numbering in other Greek ms., is not comprehensible on 
the assumption that eight preceding propositions were omitted in it, 
as they are in the Arabic; and lastly, the readings in L are tolerably 
like those of our MSS., and surprisingly tike those of B. It is also to 
be noted that, although P dates from the 10th c. only, it contains, 
according to all appearance, an ante-Theonine recension. 

Moreover there is positive evidence against certain omissions by 
the Arabians. At-Tusi omits VI. 12, but it is scarcely possible that, 
if Eutocius had not had it, he would have quoted Vi. 23 by that 
number*. This quotation of VI. 23 by Eutocius also tells against 
Ishaq who has the proposition as vi. 25. Again, Simplicius quotes VI. 
10 by that number, whereas it is VI. 13 in Ishaq ; and Pappus quotes, 
by number, XIII. 2 (Ishaq 3, 4), X.III. 4 (Ishaq 8), XIII. 16 (Ishaq 19). 

1 Heiberg in Ztitxhrift fur Math. u. Pfytit, XXIX., bilt.-litt. Ablh. p. 3 sqq. 
1 Apollonius, ed. Heiberg, vol. II. p. »i8, 3 — 5. 



8a INTRODUCTION [en. vii 

On the other hand the contraction of III. II, 12 into one proposition 
in the Arabic tells in favour of the Arabic. 

Further, the omission of certain porisms in the Arabic cannot be 
supported; for Pappus quotes the porism to XIII. 17 1 , Procius those 
to II. 4, in. 1, vii. 2\ and Simplicius that to IV. 15. 

Lastly, some propositions omitted in the Arabic are required in 
later propositions. Thus X. 13 is used in X. 18, 22, 23, 26 etc. ; X. 17 
is wanted in X. 18, 26, 36; xn. 6, 13 are required for XII. 1 1 and XII. 
15 respectively. 

It must also be remembered that some of the things which were 
properly omitted by the Arabians are omitted or marked as doubtful 
in Greek MSS. also, especially in P, and others are rightly suspected for 
other reasons (e.g. a number of alternative proofs, lemmas, and porisms, 
as well as the analyses and syntheses of XII I. 1—5). On the other 
hand, the Arabic has certain interpolations peculiar to our inferior 
MSS. (cf. the definition VI. Def. 2 and those of proportion and ordered 
proportion), 

Heiberg comes to the general conclusion that, not only is the 
Arabic tradition not to be preferred offhand to that of the Greek mss., 
but it must be regarded as inferior in authority. It is a question 
how far the differences shown in the Arabic are due to the use of 
Greek MSS. differing from those which have been most used as the 
basis of our text, and how far to the arbitrary changes made by 
the Arabians themselves. Changes of order and arbitrary omissions 
could not surprise us, in view of the preface above quoted from the 
Oxford MS. of Thabit-Ishaq, with its allusion to the many important 
and necessary things left out by Abu '1 Wafa and to the author's, 
own rearrangement of Books XII., xm. But there is evidence of 
differences due to the use by the Arabs of other Greek Mss. Heiberg' 
is able to show considerable resemblances between the Arabic text 
and the Bologna MS. b in that part of the MS. where it diverges so 
remarkably from our other MSS. (see the short description of it above, 
p. 49) ; in illustration he gives a comparison of the proofs of XII. 7 in b 
and in the Arabic respectively, and points to the omission in both of 
the proposition given in Gregory's edition as XI 38, and to a remark- 
able agreement between them as regards the order of the propositions 
of Book XII. As above stated, the remarkable divergence of b only 
affects Books xi. (at end) and XII. ; and Book xm. in b shows none 
of the transpositions and other peculiarities of the Arabic. There 
are many differences between b and the Arabic, especially in the 
definitions of Book XX, as well as in Book xm. It is therefore a 
question whether the Arabians made arbitrary changes, or the Arabic 
form is the more ancient, and b has been altered through contact 
with other MSS. Heiberg points out that the Arabians must be alone 
responsible for their definition of a prism, which only covers a prism 
with a triangular base. This could not have been Euclid's own, for 
the word prism already has the wider meaning in Archimedes, and 

1 Pippus, V. p. 436, 5. * Procius, pp. 303 — 4. 

* Ztttschrift fur Math. u. Physik, XXIX., hbt.-titt. Al.ib. p. 6«qq. 



CH, vii] EUCLID IN ARABIA 83 

Euclid himself speaks of prisms with parallelograms and polygons 
as bases (xi. 39; XII. 10). Moreover, a Greek would not have been 
likely to leave out the definitions of the " Platonic " regular solids. 

Heiberg considers that the Arabian translator had before him 
a MS. which was related to b, but diverged still further from the rest 
of our MSS. He does not think that there is evidence of the existence of 
a redaction of Books I. — X. similar to that of Books XI., XII. in b ; for 
K I am roth observes that it is the Books on solid geometry (XI. — XIIL) 
which are more remarkable than the others for omissions and shorter 
proofs, and it is a noteworthy coincidence that it is just in these 
Books that we have a divergent text in b. 

An advantage in the Arabic version is the omission of VII. Def. 10, 
although, as Iambliehus had it, it may have been deliberately omitted 
by the Arabic translator. Another advantage is the omission of the 
analyses and syntheses of XI II. I — 5 ; but again these may have been 
omitted purposely, as were evidently a number of porisms which 
are really necessary. 

One or two remarks may be added about the Arabic versions 
as compared with one another, Al-Hajjaj's object seems to have 
been less to give a faithful reflection of the original than to write 
a useful and convenient mathematical text-book. One characteristic 
of it is the careful references to earlier propositions when their results 
are used. Such specific quotations of earlier propositions are rare in 
Euclid ; but in al-Hajjaj we find not only such phrases as "by prop, 
so and so," " which was proved " or " which we showed how to do in 
prop, so and so," but also still longer phrases. Sometimes he repeats 
a construction, as in I. 44 where, instead of constructing " the parallelo- 
gram BEFG equal to the triangle C in the angle EBG which is equal 
to the angle })" and placing it in a certain position, he produces AB 
to G, making BG equal to half DE (the base of the triangle CDE in 
his figure), and on GB so constructs the parallelogram BHKG by 
I. 42 that it is equal to the triangle CDE, and its angle GBH is equal 
to the given angle. 

Secondly, al-Hajjaj, in the arithmetical books, in the theory of 
proportion, in the applications of the Pythagorean 1. 47, and generally 
where possible, illustrates the proofs by numerical examples. It is 
true, observes Klamroth, that these examples are not apparently 
separated from the commentary of an-Nairtzi, and might not there- 
fore have been due to al-Hajjaj himself; but the marginal notes to 
the Hebrew translation in Municn MS. 36 show that these additions 
were in the copy of al-Hajjaj used by the translator, for they expressly 
give these proofs in numbers as variants taken from al-Hajjaj 1 . 

These characteristics, together with al-Hajjaj 's freer formulation 
of the propositions and expansion of the proofs, constitute an in- 
telligible reason why Ishaq should have undertaken a fresh translation 
from the Greek. Klamroth calls Ishaq's version a model of a good 
translation of a mathematical text ; the introductory and transitional 



Klamruth, p. 310 ; Steiiischneidet, pp. 85 — 6. 



84 INTRODUCTION fc H - vii 

phrases are stereotyped and few in number, the technical terms are 
simply and consistently rendered, and the less formal expressions 
connect themselves as closely with the Greek as is consistent with 
intelligibility and the character of the Arabic language. Only in 
isolated cases does the formulation of definitions and enunciations 
differ to any considerable extent from the original. In general, his 
object seems to have been to get rid of difficulties and unevennesses 
in the Greek text by neat devices, while at the same time giving a 
faithful reproduction of it 1 . 

There are curious points of contact between the versions of 
al-Hajjaj and Thabit-Ishaq. For example, the definitions and 
enunciations of propositions are often word for word the same. 
Presumably this is owing to the fact that Ishaq found these de- 
finitions and enunciations already established in the schools in his 
time, where they would no doubt be leamt by heart, and refrained 
from translating them afresh, merely adopting the older version with 
some changes'. Secondly, there is remarkable agreement between 
the Arabic versions as regards the figures, which show considerable 
variations from the figures of the Greek text, especially as regards 
the letters ; this is also probably to be explained in the same way, 
all the later translators having most likely borrowed al-Hajjaj's 
adaptation of the Greek figures'. Lastly, it is remarkable that the 
version of Books XI. — XIII. in the KjfSbenhavn MS. (K), purporting 
to be by al-Hajjaj, is almost exactly the same as the Thabit-Ishaq 
version of the same Books in O. Klamroth conjectures that Ishaq 
may not have translated the Books on solid geometry at all, and that 
Thabit took them from al-Hajjaj, only making some changes in order 
to fit them to the translation of Ishaq'. 

From the facts (l) that at-TusI's edition had the same number 
of propositions (468) as al-Hajjaj's version, while Thabit- 1 shaq's had 
478, and (2) that at-Tusl has the same careful references to earlier 
propositions, Klamroth concludes that at-Tusi deliberately preferred 
al-Hajjaj's version to that of Ishaq', Heiberg, however,, points out 
(1) that at-Tusi left out VI. 12 which, if we may judge by Klamroth's 
silence, al-Hajjaj had, and (2) al-Hajjaj's version had one proposition 
less in Books 1. and in. than at-Tusl has. Besides, in a passage quoted 
by Hajl Khalfa' from at-Tusi, the latter says that "he separated the 
things which, in the approved editions, were taken from the archetype 
from the things which had been added thereto," indicating that he 
had compiled his edition from both the earlier translations'. 

There were a large number of Arabian commentaries on, or 
reproductions of, the Elements or portions thereof, which will be 

1 Klamroth, p. 390, illustrates Ish&q's method by his way of distinguishing l^op^dftur 
(to be congruent with) and lipapubfadai (to be applied to), the confusion of which by trans- 
lators was animadverted on by Savile. Ishiq avoided the confusion by using two entirely 
different words. 

' Klamroth, pp. JIO— I. ' ibid. p. 187, 

• ibid. pp. 304—5. * ibid- p. "a 74. 

• Hail Khalfa, I. p. 383. 

7 Heiberg, Zritschrift ftir Math. u.Phjrsik, XXIX., hist.-litt. Abth. pp.3, 3. 



en. vn] EUCLID IN ARABIA 85 

found fully noticed by Steinschn eider 1 . I shall mention here the 
commentators etc. referred to in the Fikrist, with a few others. 

1. Abu '1 'Abbas al-Fadl b. Hatim an-Nairlzi (born at Nairiz, 
died about 922) has already been mentioned'. His commentary 
survives, as regards Books 1. — VI., in the Codex Leidensis 399, 1, now 
edited, as to four Books, by Besthorn and Heiberg, and as regards 
Books I. — x. in the Latin translation made by Gherard of Cremona 
in -the 12th c. and now published by Curtze from a Cracow MS.* Its 
importance lies mainly in the quotations from Heron and Simplicius. 

2. Ahmad b. 'Umar al-KarablsI (date uncertain, probably 9th — 
10th c), " who was among the most distinguished geometers and 
arithmeticians'." 

3. A 1-' Abbas b. Sa'ld al-Jauhan (fl. 830) was one of the astro- 
nomical observers under al-Ma'mun, but devoted himself mostly to 
geometry. He wrote a commentary to the whole of the Elements, 
from the beginning to the end ; also the " Book of the propositions 
which he added to the first book of Euclid*." 

4. Muh. >. 'Isa Abu 'Abdallah al-Mahan! (d. between 874 and 
884) wrotej according to the Fikrist, (1) a commentary on Eucl. 
Book v., (2) "On proportion," (3) "On the 26 propositions of the 
first Book of Euclid which are proved without reductio ad absurdum*." 
The work " On proportion " survives and is probably identical with, or 
part of, the commentary on Book V. 7 He also wrote, what is not 
mentioned by the Fihrist, a commentary on Eucl. Book X., a fragment 
of which survives in a Paris MS." 

5. Abu Ja'far al- Khazin (i.e. " the treasurer " or " librarian "), one 
of the first mathematicians and astronomers of his time, was born in 
Khurasan and died between the years 961 and 971. The Fikrist 
speaks of him as having written a commentary on the whole of the 
Elements*, but only the commentary on the beginning of Book X. 
survives (in Leiden, Berlin and Paris) ; therefore either the notes on 
the rest of the Books have perished, or the Fihrist is in error 10 . The 
latter would seem more probable, for, at the end of his commentary, 
al-Khazin remarks that the rest had already been commented on by 
Sulaiman b. 'Usma (Leiden MS.) 11 or "Oqba (Surer), to be mentioned 
below. At-Khazin's method is criticised unfavourably in the preface 
to the Oxford MS. quoted by Nicoll (see p. 77 above). 

6. Abu '1 WaiS al-Buzjanl (940-997), one of the greatest 
Arabian mathematicians, wrote a commentary on the Elements, but 

1 Steinschneider, Zeiischriftfiir Mali. u. Physii, XXXI., hist.-litt. Abth, pp. 86 sqq. 
1 Steinschneider, p. 86, Fihrist (Ir. Suter), pp. [6, 67 ; Suter, Dit Maiktmaliktr ttnd 
Astrmniien Jrr Arabcr (1900), p. +J. 

' SuppitHUHtum ad Eudiftis Optra omnia, ed. Heiberg and Mtnge, Leipzig, 1899. 
1 Fikrist , pp. 16, 38 [ Sleinschneider, p. 87 ; Suter, p. 6j. 

* Fihrist, pp. 16, 15; Steinschneider, p. gg . Suter, p. 12. 

* FUrist, pp. 16, ii, 58. 

7 Suter, p. 16, note, quotes the Para MS. 1467, 16 s containing the work "on proportion" 
u the authority for this conjecture. 

1 MS. i+S7, 39° (ef. Woepcke in Mhn. pr4s. a raead. dit sriemes, XI v., i8j6\ p. 669). 

* Fikrist, p. 17. ™ Suter, p. j8, note b. " Steintchneider, p. 89. 



86 INTRODUCTION [ch. vii 

did not complete it 1 . His method is also unfavourably regarded in 
the same preface to the Oxford MS. 28a According to Haji Khalfa, he 
also wrote a book on geometrical constructions, in thirteen chapters. 
Apparently a book answering to this description was compiled by a 
gifted pupil from lectures by Abu '1 Wafa, and a Paris MS. (Anc. fonds 
169) contains a Persian translation of this work, not that of Abu '1 Wafa 
himself. An analysis of the work was given by Woepcke*, and some 
particulars will be found in Cantor'. Abu '1 Wafa also wrote a 
commentary on Diophantus, as well as a separate "book of proofs 
to the propositions which Diophantus used in his book and to what 
he (Abu '1 Wafa) employed in his commentary*." 

7. Ibn Rahawaihi al-Arjanl also commented on Eucl. Book X.* 

8. 'All b, Ahmad Abu '1-Qasim al- AntakI (d. 987) wrote a 
commentary on the whole book 1 ; part of it seems to survive (from 
the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 28 1) 7 . 

9. Sind b. 'AH Abu 't-Taiyib was a Jew who went over to 
Islam in the time of al-Ma'mun, and was received among his astro- 
nomical observers, whose head ht became* (about 830); he died after 
864. He wrote a commentary on the whole of the Elements ; " Abu 
'All saw nine books of it, and a part of the tenth 8 ." His book " On 
the Apotomae and the Medials," mentioned by the Fihrist, may be 
the same as, or part of, his commentary on Book x. 

10. Abu Yusuf Ya'qQb b, Muh. ar-Razi "wrote a commentary 
on Book X., and that an excellent one, at the instance of Ibn al- 
•Amid"." 

11. The Fihrist next mentions al-Kindi (Abu Yusuf Ya'qub b. 
Ishaq b. as-Sabbah al-Kindi, d. about 873), as the author (1) of a 
work * on the objects of Euclid's book," in which occurs the statement 
that the Elements were originally written by Apollonius, the carpenter 
(see above, p. 5 and note), (2) of a book "on the improvement of 
Euclid's work," and (3) of another "on the improvement of the 14th 
and 15th Books of Euclid." "He was the most distinguished man 
of his time, and stood alone in the knowledge of the old sciences 
collectively ; he was called ' the philosopher of the Arabians ' ; his 
writings treat of the most different branches of knowledge, as logic, 
philosophy, geometry, calculation, arithmetic, music, astronomy and 
others"." Among the other geometrical works of al-Kindi mentioned 
by the Fihrist 1 * are treatises on the closer investigation of the results 
of Archimedes concerning the measure of the diameter of a circle in 
terms of its circumference, on the construction of the figure of the two 
mean proportionals, on the approximate determination of the chords 

^r. v. T. v. pp. 118 — 156 and 309— $$$. 

* Fihrist, p. 17 ; Suter, p. ij. 
■ Fihrist, p. 17. 7 Suter, p, 64. 

• Fikrisl, p. 17, 19 j Suter, pp. 13, i+. » Fihrist, p. r;. 
" Fihrist, p. 17; Suter, p. 66. " Fihrist, p. 17, 10 — [J. 

u The mere catalogue of al-KindT's works on the various branches of science takes up 
four octavo pages {1 1— ij) of Suter's translation of the Fihrist. 

I 



ch. vii] EUCLID IN ARABIA 87 

of the circle, on the approximate determination of the chord (side) of 
the nonagon, on the division of triangles and quadrilaterals and con- 
structions for that purpose, on the manner of construction of a circle 
which is equal to the surface of a given cylinder, on the division of 
the circle, in three chapters etc. 

12. The physician Nazif b. Yumn (or Yaman) al-Qass ("the 
priest ") is mentioned by the Fikrist as having seen a Greek copy 
of Eucl. Book X. which had 40 more propositions than that which 
was in general circulation (containing 109), and having determined 
to translate it into Arabic 1 . Fragments of such a translation exist 
at Paris, Nos. 18 and 34. of the MS. 24s 7 (952, 2 Suppl. Arab, in 
Woepcke's tract); No. 18 contains "additions to some propositions 
of the 10th Book, existing in the Greek language'." Nazif must have 
died about 990*, 

13. YGhanna b. Yusuf b. al-Harith b. al-Bitriq al-Qass (d. about 
980) lectures' on the Elements and other geometrical books, made 
translations from the Greek, and wrote a tract on the " proof" of the 
case of two straight lines both meeting a third and making with it, 
on one side, two angles together less than two right angles*. Nothing 
of his appears to survive, except that a tract " on rational and irrational 
magnitudes," No. 48 in the Paris MS. just mentioned, is attributed 
to him. 

14. Abu Muh. al- Hasan b. 'Ubaidallah b. Sulatman b. Wahb 
(d. 901) was a geometer of distinction, who wrote works under the 
two distinct titles " A commentary on the difficult parts of the work 
of Euclid " and " The Book on Proportion'." Suter thinks that an- 
other reading is possible in the case of the second title, and that it 
may refer to the Euclidean work " on the divisions (of figures)*." 

15. Qusta b. Luqi al-Ba'labakkl (d. about 912), a physician, 
philosopher, astronomer, mathematician and translator, wrote " on the 
difficult passages of Euclid's book" and "on the solution of arith- 
metical problems from the third book of Euclid 7 "; also an "intro- 
duction to geometry," in the form of question and answer". 

16. Thabit b. Qurra (826-90 1), besides translating some parts 
of Archimedes and Books V. — VII. of the Conks of Apollonius, and 
revising Ishaq's translation of Euclid's Elements, also revised the trans- 
lation of the Data by the same Ishaq and the book On divisions of 

figures translated by an anonymous writer. We are told also 
that he wrote the following works : (1) On the Premisses (Axioms, 
Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On 
the propositions and questions which arise when two straight lines 
are cut by a third (or on the "proof" of Euclid's famous postulate). 
The last tract is extant in the MS. discovered by Woepcke (Paris 
2 457> 3*°)- He is also credited with "an excellent work" in the 
shape of an " Introduction to the Book of Euclid," a treatise on 

1 Fihriit, pp. 16, 1 j. 

* Woepcke, Mtm. pris. i I'acad. da scittua, XIV. pp. 666, 668. 

* Suter, p. 68. * FiAmt, p. 38 ; Suter. p. 60. 
1 Fikrist, p. 16, rad Suter' 5 Dote, p. 60. * Suter, p. tit, note 13, 

' Fikritt, p. 43, * Fihriit, p. +3 ; Soter, p. 41. 



88 INTRODUCTION [ch. vii 

Geometry dedicated to Ismail b, Bulbul, a Compendium of Geometry, 
and a large number of other works for the titles of which reference 
may be made to Suter, who also gives particulars as to which are 
extant 1 . 

17. Abu Sa'ld Sinan b. T habit b, Qurra, the son of the translator 
of Euclid, followed in his father's footsteps as geometer, astronomer 

and physician. He wrote an "improvement of the book of on 

the Elements of Geometry, in which he made various additions to the 
original." It is natural to conjecture that Euclid is the name missing 
in this description (by Ibn abl Usaibi'a); Casiri has the name Aqaton 1 . 
The latest editor of the Ta'rikk al-Hukamd, however, makes the name 
to be Iflaton (= Plato), and he refers to the statement by the Fikrist 
and Ibn al-Qiftl attributing to Plato a work on the Elements of 
Geometry translated by Qusta. It is just possible, therefore, that at 
the time of Qusta the Arabs were acquainted with a book on the 
Elements of Geometry translated from the Greek, which they attri- 
buted to Plato*. Sinan died in 94.3. 

18. Abu Sahl Wijan (or Waijan) b. Rustam al-Kuhi (ft 988), 
born at Kuh in Tabaristan, a distinguished geometer and astronomer, 
wrote, according to the Fikrist, a " Book of the Elements" after that 
of Euclid*; the 1st and 2nd Books survive at Cairo, and a part of 
the 3rd Book at Berlin (5922)'. He wrote also a number of other 
geometrical works : Additions to the 2nd Book of Archimedes on 
the Sphere and Cylinder (extant at Paris, at Leiden, and in the India 
Office), On the finding of the side of a heptagon in a circle (India 
Office and Cairo), On two mean proportionals (India Office), which 
last may be only a part of the Additions to Archimedes' On the Sphere 
and Cylinder, etc. 

19. Abu Nasr Muh. b. Muh. b. Tarkhan b. Uzlag al-Farabl 
(870-950) wrote a commentary on the difficulties of the introductory 
matter to Books I. and V. s This appears £0 survive in the Hebrew 
translation which is, with probability, attributed to Moses b. Tibbon'. 

20. Abu 'All al-Hasan b. al- Hasan b, al-Haitham (about 965- 
1039), known by the name Ibn al-Haitham or Abu 'AHal-Basri, was a 
man of great powers and knowledge, and no one of his time approached 
him in the field of mathematical science. He wrote several works on 
Euclid the titles of which, as translated by Woepcke from Usaibi'a, 
are as follows 5 : 

1. Commentary and abridgment of the Etements. 

2. Collection of the Elements of Geometry and Arithmetic, 
drawn from the treatises of Euclid and Apollonius. 

3. Collection of the Elements of the Calculus deduced from 
the principles laid down by Euclid in his Elements. 

1 Suter, pp. 34—8. 

1 Fikritt (ed. Suter), p. 59, note 131 ; Suter, p. ji, note b. 

* See Suter in Bibliathtca Mathematical Jv a> 1903-4, pp. 396 — 7, review of JuIluk 
Lippert's Ihn al-Qifti. Ta^rith al-hiikamd, Leipzig, 1903. 

■ Fikritt, p. 40. • Suter, p. <n, 

' Suter, p. 55. ' Steinschneider, p. 91. 

■ Steinscbneider, pp. 91 — 3. 



CK. vn] EUCLID IN ARABIA 89 

4. Treatise on " measure " after the manner of Euclid's 
Elements, 

5. Memoir on the solution of the difficulties in Book I. 

6. Memoir for the solution of a doubt about Euclid, relative 
to Book v, 

7. Memoir on the solution of a doubt about the stereometric 
portion. 

8. Memoir on the solution of a doubt about Book XII. 

9. Memoir on the division of the two magnitudes mentioned 
in X. r (the theorem of exhaustion). 

10. Commentary on the definitions in the work of Euclid 
(where Steinschneider thinks that some more general expression 
should be substituted for "definitions"). 

The last-named work (which Suter calls a commentary on the 
Postulates of Euclid) survives in an Oxford ms. (Catal. MSS. orient. 
I. 908) and in Algiers (1446, i"). 

A Leiden MS. (966) contains his Commentary "on the difficult 
places " up to Book v. We do not know whether in this commentary, 
which the author intended to form, with the commentary on the 
Musadarat, a sort of complete commentary, he had collected the 
separate memoirs on certain doubts and difficult passages mentioned 
in the above list 

A commentary on Book V. and following Books found in a 
Bodleian MS. (Catal. II. p. 262) with the title " Commentary on Euclid 
and solution of his difficulties " is attributed to b. Haitham ; this might 
be a continuation of the Leiden US. 

The memoir on X. 1 appears to survive at St Petersburg, MS. de 
l'lnstitut des langues orient. 192, 5° (Rosen, Catal. p. 125). 

21. Ibn Sina, known as Avicenna (980-1037), wrote a Com- 
pendium of Euclid, preserved in a Leiden MS. No. 1445, and forming 
the geometrical portion of an encyclopaedic work embracing Logic, 
Mathematics, Physics and Metaphysics 1 . 

22. Ahmad b. al-Husain al-Ahwazt al-Katib wrote a com-: 
mentary on Book X., a fragment of which (some 10 pages) is to be 
found at Leiden (970), Berlin (5923) and Paris (2467, 1 8°)*. 

25. Naslraddln at-TusT (1 201-1274) who, as we have seen, 
brought out a Euclid in two forms, wrote : 

1. A treatise on the postulates of Euclid (Paris, 2467, 5 ). 

2. A treatise on the 5th postulate, perhaps only a part of 
the foregoing (Berlin, 5942, Paris, 2467, 6°). 

3. Principles of Geometry taken from Euclid, perhaps 
identical with No. I above (Florence, Pal. 298). 

4. 105 problems out of the Elements (Cairo). He also edited 
the Data (Berlin, Florence, Oxford etc.)*, 

24. Muh. b. Ashraf Shamsaddin as-Samarqandi (fl. 1276) wrote 
"Fundamental Propositions, being elucidations of 35 selected proposi- 

' Sleinschneider, p. 91 ; Suter, p. 89. - Suter, p. 57. 

' Suter, pp. ijo — i. 



go INTRODUCTION [ch. vii 

tions of the first Books of Euclid," which are extant at Gotha (1496 
and 1497), Oxford (Catal. I. 967, 2% and Brit Mus. 1 . 

25. Musa b. Muh. b. Mahmud, known as Qadizade ar-Rumi (i.e. 
the son of the judge from Asia Minor), who died between 1436 and 
1446, wrote a commentary on the "Fundamental Propositions" just 
mentioned, of which many MSS. are extant 1 . It contained biographical 
statements about Euclid alluded to above (p. 5. note), 

26. Abu Da'Gd Sulaiman b. 'Uqba, a contemporary of al-Khazin 
(see above, No. 5), wrote a commentary on the second half of Book X., 
which is, at least partly, extant at Leiden (974) under the title "On 
the binomials and apotomae found in the loth Book of Euclid*." 

27. The Codex Leidensis 399, 1 containing al-yajjaj's transla- 
tion of Books I. — VL is said to contain glosses to it by Sa id b Mas'ud 
b. al-Qass, apparently identical with Abu Nasr Gars al-Na'ma, son of 
the physician Mas'ud b. al-Qass al-Bagdadl, who lived in the time of 
the last Caliph al-Musta'sim {d. 1258)'. 

28. Abu Muhammad b. AbdalbaqT al-Bagdadl al-Faradl (d. 
1141, at the age of over 70 years) is stated in the Ta'rikk al-Hukama 
to have written an excellent commentary on Book X. of the Elements, 
in which he gave numerical examples of the propositions'. This is 
published in Curtze's edition of an-Nairizi where it occupies pages 
252 — 386". 

29. Yahya b. Muh b. 'Abdan b. 'Abdalwahid, known by the 
name of Ibn al-Lubudi (1 210-1268), wrote a Compendium of Euclid, 
and a short presentation of the postulates*. 

30. Abu 'Abdallah Muh. b. Mu'adh al-Jayyanl wrote a com- 
mentary on Eucl. Book V. which survives at Algiers (1446, 3°)'. 

31. Abu Nasr Mansur b. 'All b. 'Iraq wrote, at the instance of 
Muh. b. Ahmad Abu 'r-Raihan al-Blruni (973-1048), a tract "on 
a doubtful (difficult) passage in Eucl. Book XIII." (Berlin, 5925)*, 

1 Suter, p. IJ7. ' ibid. p. 175. * ibid. p. $6. 

1 ibid. pp. 153— 4, nj. 

* Gartx, p. 14 ; Steinschneider, pp- 94 — 5. 

* Suter in Bibliothtca Mathtmatua, IV,, 19031 PP' »£* »9S 1 Suter has also an article an 
its contents, Biblintheta Mathtmatica, VII,, 1906-7, pp. 134 — ajl. 

* Steinsctaneider, p. 94 ; Suter, p. 146. 

* Suter, Nacktrige und Btrichtigurtgen, in Abk&ndlungt* »*' Goth, dtr matk, JVisstn- 
Khafltn, xiv., 1901, p. 170, 

* Suter, p. 8], and Nathtrage, p. 173. 



CHAPTER VIII. 

PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS. 

Cicero is the first Latin author to mention Euclid 1 ; but it is not 
likely that in Cicero's time Euclid had been translated into Latin or 
was studied to any considerable extent by the Romans ; for, as Cicero 
says in another place ', while geometry was held in high honour 
among the Greeks, so that nothing was more brilliant than their 
mathematicians, the Romans limited its scope by having regard only 
to its utility for measurements and calculations. How very little 
theoretical geometry satisfied the Roman agrimensores is evidenced 
by the work of Balbus de mensuris', where some of the definitions of 
Eucl. Book i, are given. Again, the extracts from the Elements found 
in the fragment attributed to Censorinus (fl. 238 A.D )• are confined to 
the definitions, postulates, and common notions. But by degrees the 
Elements passed even among the Romans into the curriculum of a 
liberal education ; for Martianus Capella speaks of the effect of the 
enunciation of the proposition " how to construct an equilateral 
triangle on a given straight line " among a company of philosophers, 
who, recognising the first proposition of the Elements, straightway 
break out into encomiums on Euclid". But the Elements were then 
(c. 470 A.D.) doubtless read in Greek ; for what Martianus Capella 
gives* was drawn from a Greek source, as is shown by the occurrence 
of Greek words and by the wrong translation of I. def. 1 (" punctum 
vero est cuius pars nihil est"). Martianus may, it is true, have 
quoted, not from Euclid himself, but from Heron or some other ancient 
source. 

But it is clear from a certain palimpsest at Verona that some 
scholar had already attempted to translate the Elements into Latin. 
This palimpsest 7 has part of the " Moral reflections on the Book of 
Job " by Pope Gregory the Great written in a hand of the 9th c. above 
certain fragments which in the opinion of the best judges date from 
the 4th c. Among these are fragments of Vergil and of Livy, as well 
as a geometrical fragment which purports to be taken from the I4tn 
and 1 5th Books of Euclid, As a matter of fact it is from Books XIL 
and XIII. and is of the nature of a free rendering, or rather a new 

1 Di tratort III. 131. ' Tusc. I. j. 

* Gromalici Vtttra, 1. 97 sij. (eri. F. Blume, K. Lachmann and A. Rudorff, Bet] in, 
1848, 1851). 

* Censorinus, ed. Hultsch, pp. 60—3. 

* Martianus Capella, VI. 714. ■ ibid. VI, 708 sq. 
7 Cf. Canlot, ],, p. s6j. 



93 INTRODUCTION Jen. vm 

arrangement, of Euclid with the propositions in different order 1 . The 
mh. was evidently the translator's own copy, because some words are 
struck out and replaced by synonyms. We do not know whether the 
translator completed the translation of the whole, or in what relation 
his version stood to our other sources. 

Magnus Aurelius Cassiodorus (b. about 475 A.D.) in the geometrical 
part of his encyclopaedia De artibus ac disciplines liberalium literarum 
says that geometry was represented among the Greeks by Euclid, 
Apollonius, Archimedes, and others, " of whom Euclid was given us 
translated into the Latin language by the same great man Boethius ." ; 
also in his collection of letters* is a letter from Theodoric to Boethius 
containing the words, "for in your translations ... Nicomachus the 
arithmetician, and Euclid the geometer, are heard in the Ausonian 
tongue." The so-called Geometry of Boethius which has come down 
to us by no means constitutes a translation of Euclid. The MSS. 
variously give five, four, three or two Books, but they represent only 
two distinct compilations, one normally in five Books and the other 
in two. Even the latter, which was edited by Friedlein, is not 
genuine*, but appears to have been put together in the 11 th c, from 
various sources. It begins with the definitions of Eucl. I., and in these 
are traces of perfectly correct readings which are not found even in 
the MSS. of the 10th c, but which can be traced in Prod us and other 
ancient sources ; then come the Postulates (five only), the Axioms 
(three only), and after these some definitions of Eucl. 1 1., ill., IV. 
Next come the enunciations of Eucl. I., of ten propositions of Book II., 
and of some from Books HI., IV., but always without proofs ; there 
follows an extraordinary passage which indicates that the author will 
now give something of his own in elucidation of Euclid, though what 
follows is a literal translation of the proofs of Eucl. I. 1 — 3. This 
latter passage, although it affords a strong argument against the 
genuineness of this part of the work, shows that the Pseudoboethius 
had a Latin translation of Euclid from which he extracted the three 
propositions. 

Curtze has reproduced, in the preface to his edition of the trans- 
lation by Gherard of Cremona of an-Nairizt's Arabic commentary on 
Euclid, some interesting fragments of a translation of Euclid taken 
from a Munich MS. of the 10th c. They are on two leaves used 
for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 
2° 757) and consist of portions of Eucl. I. 37, 38 and II. 8, translated 
literally word for word from the Greek text. The translator seems to 
have been an Italian (cf. the words "capitolonono" used for the ninth 
prop, of Book II.) who knew very little Greek and had moreover little 
mathematical knowledge. For example, he translates the capital letters 
denoting points in figures as if they were numerals : thus to. A Br, 

1 The fragment was deciphered by W. Studernund, who communicated his results to 
Cantor. 

* Cassiodorus, Variat, I. 4;, p. +0, 13 cd. Mommsen. 

* See especial Ijr Weissenbom in Abhandltmgen tur Gtsti. d. Math. 11. p. 185 so,.; 
Heiberg in Phihlogus, XLI1I. p. 507 sq. ; Cantor, 13, p. 580 so,. 



ch. vni] TRANSLATIONS AND EDITIONS 93 

AEZ is translated "que primo secundo et tertio quarto quinto et 
sept i mo," T becomes "tricentissimo " and so on. The Greek MS. which 
he used was evidently written in uncials, for AEZ8 becomes in one 
place " quod autem septimo nono," showing that he mistook AE for 
the particle Si, and xal 6 2TU is rendered "sicut tricentissimo et 
quadringentissimo," showing that the letters must have been written 
KAIOCTU. 

The date of the Englishman Athelhard (^Ethelhard) is approxi- 
mately fixed by some remarks in his work Perdifficites Quaestiones 
Naturaks which, on the ground of the personal allusions they contain, 
must be assigned to the first thirty years of the 12th c. 1 He wrote a 
number of philosophical works. Little is known about his life. He 
is said to have studied at Tours and Lao 11, and to have lectured at the 
latter school. He travelled to Spain, Greece, Asia Minor and Egypt, 
and acquired a knowledge of Arabic, which enabled him to translate 
from the Arabic into Latin, among other works, the Elements of 
Euclid. The date of this translation must be put at about 1120. 
MSS. purporting to contain Atheihard's version are extant in the 
British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 
47 and Ball. Coll. 257 of 12th c), Niirnberg (Johannes Regiomontanus' 
copy) and Erfurt. 

Among the very numerous works of Gherard of Cremona (I 1 14 — 
1 1 87) are mentioned translations of " 1 5 Books of Euclid " and of the 
Data*. Till recently this translation of the Elements was supposed to 
be lost; but Axel Anthon Bjornbo has succeeded (1904) in discovering 
a translation from the Arabic which is different from the two others 
known to us (those by Athelhard and Campanus respectively), and 
which he, on grounds apparently convincing, holds to be Gherard's. 
Already in 190 1 Bjornbo had found Books X.— XV. of this translation 
in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)*; but three 
years later he had traced three MSS. containing the whole of the same 
translation at Paris (Cod. Paris. 7216, 15th a), Baulogne-sur-Mer 
(Cod. Bononiens. 196, 14th a), and Bruges (Cod, Brugens. 521, 14th c), 
and another at Oxford (Cod. Digby [74, end of 12th c.) containing a 
fragment, XI. 2 to XIV. The occurrence of Greek words in this 
translation such as rombus, romboides (where Athelhard keeps the 
Arabic terms), ambligonius, ortAogonius, gnomo, fyramis etc., show 
that the translation is independent of Atheihard's. Gherard appears 
to have had before him an old translation of Euclid from the Greek 
which Athelhard also often followed, especially in his terminology, 
using it however in a very different manner. Again, there are some 
Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not 
use, but which is found in almost all the translations that are with 
certainty attributed to Gherard of Cremona; there occurs also the 

1 Cautor, Guti. d. Math. („ p. 006. 

1 Boncompagni, Delia vita t Mil ogcrt di Gherorih Crcmmusc, Rome, 1 851, p. 5. 

* Described in an »ppendix to Sludun iibir Mttuloes'' SpAarik [Abliaiidlungtn lur 
Gtithithtt dtr motktmaHichttt Wisumchafttn, xtv., 1901). 

* See Biblitthtca Matktmalka, vi„ 1903-6, pp. 141 — 6, 



94 INTRODUCTION [cb. vjii 

expression "superficies equidistantium laterum et rcctorum angulorum," 
found also in Gherard's translation of an-Nairlzl, where Athelhard says 
"parallelogrammum rectangulum." The translation is much clearer 
than Athelhard's: it is neither abbreviated nor "edited" as A the! hard's 
appears to have been ; it is a word-for-word translation of an Arabic 
MS. containing a revised and critical edition of Thabit's version. It 
contains several notes quoted from Thabit himself ( Tftebit dixit), e.g. 
about alternative proofs etc. which Thabit found " in another Greek 
MS.," and is therefore a further testimony to Thabit's critical treatment 
of the text after Greek MSS. The new editor also added critical 
remarks of his own, e.g. on other proofs which he found in other 
Arabic versions, but not in the Greek : whence it is clear that he 
compared the Thabit version before him with other versions as care- 
fully as Thabit collated the Greek mss. Lastly, the new editor speaks 
of "Thebit qui transtulit hunc librum in arabicam linguam" and of 
"translatio Thebit," which may tend to confirm the statement of al-Qiftl 
who credited Thabit with an independent translation, and not (as the 
Fihrist does) with a mere improvement of the version of Ishaq b, 
Hunain. 

Gherard's translation of the Arabic commentary of an-Nairlzi on 
the first ten Books of the Elements was discovered by Maximilian 
Curtze in a MS. at Cracow and published as a supplementary volume 
to Heiberg and Menge's Euclid 1 : it will often be referred to in this 
work. 

Next in chronological order comes Johannes Campanus (Campano) 
of Novara. He is mentioned by Roger Bacon (12 14-1 294) as a 
prominent mathematician of his time 1 , and this indication of his date 
is confirmed by the fact that he was chaplain to Pope Urban IV, who 
was Pope from 1261 to 1281". His most important achievement was 
his 'edition of the Elements including the two Books XIV. and XV. 
which are not Euclid's. The sources of Athelhard's and Campanus' 
translations, and the relation between them, have been the subject of 
much discussion, which does not seem to have led as yet to any 
definite conclusion. Cantor (Hi, p. 91) gives references* and some 
particulars. It appears that there is a Ms. at Munich (Cod. lat. Mon. 
13021} written by Sigboto in the 12th c. at Priifning near Regensburg, 
and denoted by Curtze by the letter R, which contains the enunciations 
of part of Euclid. The Munich MSS. of Athelhard and Campanus' 
translations have many enunciations textually identical with those in 
R, so that the source of all three must, for these enunciations, have 

1 Anaritii in decern libros prism Etemenierum Eucfidis Comnuntarii ex interpretation* 
Gherardi Cremenensis in codicc Cracaviensi 569 strvata edidit M ax i mi Nanus Curtze, Leipzig 
(Teubner), 1899. 

* Cantor, iij, p. 88. 

* Tiraboschi, Storia delta tetteratura italiana, IV, 145 — 160, 

4 H. Weissenborn in Zeitichrift fiir Math. u. Physii, XXV., Supplement, pp. i+j— 166, 
and in his monograph. Die Ubersttiungen des Euktid dureh Campano und Zamberti (188?); 
Max. Curtze in Philelogisc&e Rundschau (iSSi), I. pp. $13—950, and in Jakrtsherieht iiitr 
die Fertschritti det tlaaischen Alterthumfwissemchafl t XL. (188+, in.) pp. 19 — »' ; Heiberg 
in ZeitscArift fiir Math. u. Pfyrik, xxxv., bist.-litt. Ablh., pp. 48 — j8 and pp. Si— 6. 



en. vm] TRANSLATIONS AND EDITIONS 95 

been the same; in others Athelhard and Campanus diverge com- 
pletely from R, which in these places follows the Greek text and is 
therefore genuine and authoritative. In the 32nd definition occurs the 
word "elinuam," the Arabic term for " rhombus," and throughout the 
translation are a number of Arabic figures. But R was not translated 
from the Arabic, as is shown by (among other things) its close 
resemblance to the translation from Euclid given on pp. 377 sqq. of 
the Gromatici Ve teres and to the so-called geometry of Boethius. The 
explanation of the Arabic figures and the word " elinuam " in Def. 32 
appears to be that R was a late copy of an earlier original with 
corruptions introduced in many places ; thus in Def. 32 a part of the 
text was completely lost and was supplied by some intelligent copyist 
who inserted the word "elinuam," which was known to him, and also 
the Arabic figures. Thus Athelhard certainly was not the first to 
translate Euclid into Latin ; there must have been in existence before 
the nth c, a Latin translation which was the common source of R, 
the passage in the Gromatici, and " Boethius." As in the two latter 
there occur the proofs as well as the enunciations of 1. 1 — 3, it is 
possible that this translation originally contained the proofs also. 
Athelhard must have had before him this translation of the 
enunciations, as well as the Arabic source from which he obtained his 
proofs. That some sort of translation, or at least fragments of one, 
were available before Athelhard's time even in England is indicated 
by some old English verses 1 : 

"The clerk Euclide on this wyse hit fonde 
Thys craft of gemetry yn Egypte londe 
Yn Egypte he tawghte hyt fill wyde, 
In dyvers londe on every syde. 
Mony erys after warde y understonde 
Yer that the Craft com ynto thys londe. 
Thys Craft Coin into England, as y yow say, 
Yn tyme of good kyng Adelstone's day," 

which would put the introduction of Euclid into England as far back 
as 924-940 A.D. 

We now come to the relation between Athelhard and Campanus. 
That their translations were not independent, as Weissenborn would 
have us believe, is clear from the fact that in all MSS. and editions, 
apart from orthographical differences and such small differences as 
are bound to arise when MSS. are copied by persons with some 
knowledge of the subject-matter, the definitions, postulates, axioms, 
and the 364 enunciations are word for word identical in Athelhard 
and Campanus; and this is the case not only where both have the 
same text as R but where they diverge from it. Hence it would seem 
that Campanus used Athelhard's translation and only developed the 
proofs by means of another redaction of the Arabian Euclid. It is 
true that the difference between the proofs of the propositions in the 
two translations is considerable; Athelhard's are short and com- 

1 Quoted by HidUwdl in Kara Mathtmatua (p. j6 note) from us. Bib. Reg. Mm. Brit. 
17 A. 1. f. ** — 3. 



96 INTRODUCTION [ch. viii 

pressed, Campari us' clearer and more complete, following the Greek 
text more closely, though still at some distance. Further, the 
arrangement in the two is different; in Athelhard the proofs regularly 
precede the enunciations, Campanus follows the usual order. It is a 
question how far the differences in the proofs, and certain additions in 
each, are due to the two translators themselves or go back to Arabic 
originals. The latter supposition seems to Curtze and Cantor the 
more probable one. Curtze's general view of the relation of Campanus 
to Athelhard is to the effect that A thel hard's translation was gradually 
altered, from the form in which it appears in the two Erfurt MSS. 
described by Weissenborn, by successive copyists and commentators 
who had Arabic originals before them, until it took the form which 
Campanus gave it and in which it was published. In support of this 
view Curtze refers to Regiomontanus' copy of the Athelhard-Campanus 
translation. In Regiomontanus' own preface the title is given, and 
this attributes the translation to Athelhard ; but, while this copy 
agrees almost exactly with Athelhard in Book I., yet, in places where 
Campanus is more lengthy, it has similar additions, and in the later 
Books, especially from Book HI. onwards, agrees absolutely with 
Campanus; Regiomontanus, too, himself implies that, though the 
translation was Athelhard's, Campanus had revised it; for he has 
notes in the margin such as the following, "Campani est hec," "dubito 
an demonstret hie Campanus " etc. 

We come now to the printed editions of the whole or of portions 
of the Elements, This is not the place for a complete bibliography, 
such as Riccardi has attempted in his valuable memoir issued in five 
parts between 1 88/ and 1893, which makes a large book in itself 1 . 
I shall confine myself to saying something of the most noteworthy 
translations and editions. It will be convenient to give first the Latin 
translations which preceded the publication of the editio prineeps of 
the Greek text in 1533, next the most important editions of the Greek 
text itself, and after them the most important translations arranged 
according to date of first appearance and languages, first the Latin 
translations after 1533, then the Italian, German, French and English 
translations in order. 

It may be added here that the first allusion, in the West, to the 
Greek text as still extant is found in Boccaccio's commentary on the 
Divina Commedia of Dante*. Next Johannes Regiomontanus, who 
intended to publish the Elements after the version of Campanus, but 
with the latter's mistakes corrected, saw in Italy (doubtless when 
staying with his friend Bessarion) some Greek MSS. and noticed how 
far they differed from the Latin version (see a letter of his written in 
the year 1471 to Christian Roder of Hamburg)*. 

1 Saggie di una Bibliografia Euttidta, memoria del Prof. Pietro Riccardi (Bologna, 
1887, 18M8, 1890, 1693). 

* 1. p. 404. 

8 Published in C. T. de Murr's Memorabilia BibikHhtcarum Mffrimbergmnttm, Part 1. 
p. 190 scjh- 



ch. viu] TRANSLATIONS AND EDITIONS 97 

I. Latin translations prior to 1533. 

1482. In this year appeared the first printed edition of Euclid, 
which was also the first printed mathematical book of any import- 
ance. This was printed at Venice by Erhard Ratdolt and contained 
Campanus' translation 1 , Ratdolt belonged to a family of artists at 
Augsburg, where he was born about 1443. Having learnt the trade 
of printing at home, he went in 1475 to Venice, and founded there a 
famous printing house which he managed for 1 1 years, after which he 
returned to Augsburg and continued to print important books until 
1516. He is said to have died in 1528. Kastner 1 gives a short 
description of this first edition of Euclid and quotes the dedication to 
Prince Mocenigo of Venice which occupies the page opposite to the 
first page of text. The book has a margin of 2i inches, and in this 
margin are placed the figures of the propositions. Ratdolt says in 
his dedication that at that time, although books by ancient and 
modern authors were printed every day in Venice, little or nothing 
mathematical had appeared : a fact which he puts down to the diffi- 
culty involved by the figures, which no one had up to that time 
succeeded in printing. He adds that after much labour he had 
discovered a method by which figures could be produced as easily as 
letters*. Experts are in doubt as to the nature of Ratdolt's discovery. 
Was it a method of making figures up out of separate parts of figures, 
straight or curved lines, put together as letters are put together to 
make words? In a life of Joh. Gottlob Immanuel Breitkopf, a con- 
temporary of Kastner's own, this member of the great house of 
Breitkopf is credited with this particular discovery. Experts in that 
same house expressed the opinion that Ratdolt's figures were wood- 
cuts, while the letters denoting points in the figures were like the 
other letters in the text ; yet it was with carved wooden blocks that 
printing began. If Ratdolt was the first to print geometrical figures, 
it was not long before an emulator arose ; for in the very same year 
Mattheus Cordonis of Windischgratz employed woodcut mathematical 
figures in printing Oresme's De latitudintbus*. How eagerly the 
opportunity of spreading geometrical knowledge was seized upon is 
proved by the number of editions which followed in the next few 
years. Even the year 1482 saw two forms of the book, though they 
only differ in the first sheet Another edition came out in i486 
(Vlmae, apud lo. Regeruni) and another in 149 1 {Vincentiae per 

1 Curtie (An-Nairi/i, p. xiji) repioduces the heading of the first page of the text as 
follows (there is no title-page) : PreclarifTimu opus elemento^t- Euclidis megaref is vna cu 
comentis Camparii pfpicaciffimi in arte geometria incipit iclicit 1 , after which the definitions 
begin at once. Other copies have the shorter heading ; Preclarissimus liber elernentorum 
Euclidis perspicacissimi : in artem Geometric incipit quam foeltrissime* At the end stands 
the following : Q Opus elcmentoru euclidis megarenf is in geometria arte Jn id quoq) Camparu 
pfpicacifTimi Cdmentaliones finiut. Erhardus ratdolt Augustensis impreflbr Ic-lertiiritnus . 
venetijs imprefTit . Anno falutis • M.cccc.Ixxxij - Octauis . Calefi . Juh . Lector . Vale. 

• Kastner, Gtuhhhttdtr Mathematii, i, p. J89 sqq. See also Weissenborn, DU (ffenctw- 
wtgen dts Euktid durtk Catttpane und Zanthtrti, pp. 1 — 7. 

- "Mea industria non sine maxiroo labore effect vt qua facilitate Litterarum elementa 
imprimuntur ea etiam geometrice figure cunficerentur. " 

' Cume in Ztifschrift fur Math. u. fhysii, xx... hist.-litt. Abth. p. 58. 



98 INTRODUCTION [ch. vm 

Leonardum de Basika et Gttlielmum de Papia), but without the dedi- 
cation to Mocenigo who had died in the meantime (1485). If Cam- 
pan us added anything of his own, his additions are at all events not 
distinguished by any difference of type or otherwise; the enunciations 
are in large type, and the rest is printed continuously in smaller type. 
There are no superscriptions to particular passages such as Euclides 
ex Campano, Campanus, Campani additio, or Campani annotatio, which 
are found for the first time in the Paris edition of 15 16 giving 
both Campanus' version and that of Zamberti (presently to be men- 
tioned). 

1 50 1. G. Valla included in his encyclopaedic work De expetendis 
et fugiendis rebus published in this year at Venice (in aedibus Aldi 
Romani) a number of propositions with proofs and scholia translated 
from a Greek MS. which was once in his possession (cod, Mutin. Ill 
B, 4 of the 15th c.). 

1505. In this year Bartolomeo Zamberti (Zambertus) brought out 
at Venice the first translation, from the Greek text, of the whole of the 
Elements. From the title 1 , as well as from his prefaces to the Catoptrica 
and Data, with their allusions to previous translators " who take some 
things out of authors, omit some, and change some," or " to that most 
barbarous translator " who filled a volume purporting to be Euclid's 
"with extraordinary scarecrows, nightmares and phantasies," one object 
of Zamberti's translation is clear. His animus against Campanus 
appears also in a number of notes, e.g. when he condemns the terms 
"helmuain" and " helmuariphe " used by Campanus as barbarous, 
un- Latin etc., and when he is roused to wrath by Campanus' unfortu- 
nate mistranslation of V. Def. 5. He does not seem to have had the 
penetration to see that Campanus was translating from an Arabic, 
and not from a Greek, text Zamberti tells us that he spent 
seven years over his translation of the thirteen Books of the 
Elements. As he seems to have been born in 1473, and the Elements 
were printed as early as 1 500, though the complete work (including the 
Pkaenomena, Optica, Catoptrica, Data etc.) has the date 1505 at the 
end, he must have translated Euclid before the age of 30. Heiberg 
has not been able to identify the Ms, of the Elements which Zamberti 
used ; but it is clear that it belonged to the worse class of mss., since 
it contains most of the interpolations of the Theonine variety. Zam- 
berti, as his title shows, attributed the proofs to Theon. 

1 509. As a counterblast to Zamberti, Luca Faciuolo brought out 
an edition of Euclid, apparently at the expense of Ratdolt, at Venice 
{per Paganinunt de Pagattinis), in which he set himself to vindicate 
Campanus. The title-page of this now very rare edition' begins thus : 
"The works of Euclid of Megara, a most acute philosopher and without 

1 The title begins thus: "Euclidis megaresis philosophi platonicj matbenuticanun 
dLscipLinaruin Janitoris : II a bent in hoc volumine quicunque ad mathematicam aubstantiam 
aspirant ; elemenLorum libros xiij cum exposition Theonis io&ignis mathematici. qui bos 
multa quae deerant ex lectione graeca sumpla addLta sunt nee non plurima perueru et 
praepostere : vol u la in Campani interpretatione ; ordinate digesta et castigata sunt etc." 
For a description of the book see Weissenborn, p. 12 fiqq- 

tt See Weissenborn, p. 30 sqq. 



CB. vih] TRANSLATIONS AND EDITIONS 99 

question the chief of all mathematicians, translated by Campanus their 
most faithful interpreter'' It proceeds to say that the translation had 
been, through the fault of copyists, so spoiled and deformed that it 
could scarcely be recognised as Euclid, Luca Faciuolo accordingly 
has polished and emended it with the most critical judgment, has 
corrected 129 figures wrongly drawn and added others, besides supply- 
ing short explanations of difficult passages. It is added that Scipio 
Vegius of Milan, distinguished for his knowledge " of both languages" 
(i.e. of course Latin and Greek), as well as in medicine and the more 
sublime studies, had helped to make the edition more perfect. Though 
Zamberti is not once mentioned, this latter remark must have refer- 
ence to Zamberti's statement that his translation was from the Greek 
text ; and no doubt Zamberti is aimed at in the wish of Paciuolo's 
"that others too would seek to acquire knowledge instead of merely 
showing off, or that they would not try to make a market of the 
things of which they are ignorant, as it were (selling) smoke'." 
Weissenborn observes that, while there are many trivialities in Paci- 
uolo's notes, they contain some useful and practical hints and explana- 
tions of terms, besides some new proofs which of course are not 
difficult if one takes the liberty, as Paciuolo does, of diverging from 
Euclid's order and assuming for the proof of a proposition results not 
arrived at till later. Two not inapt terms are used in this edition to 
describe the figures of HI. 7, 8, the former of which is called the 
goose's foot (pes anseris), the second the peacock's tail (cauda pavonis) 
Paciuolo as the castigator of Campanus' translation, as he calls himself, 
failed to correct the mistranslation of v, Def. 5 s , Before the fifth 
Book he inserted a discourse which he gave at Venice on the 
15th August, 1508, in S. Bartholomew's Church, before a select 
audience of 500, as an introduction to his elucidation of that Book. 
1516. The first of the editions giving Campanus' and Zamberti's 
translations in conjunction was brought out at Paris (in officina Henrici 
Stephani e regiene scholae Decretorum). The idea that only the enun- 
ciations were Euclid's, and that Campanus was the author of the proofs 
in his translation, while Theon was the author of the proofs in the Greek 
text, reappears in the title of this edition ; and the enunciations of the 
added Books XIV., XV. are also attributed to Euclid, Hypsicles being 
credited with the proofs 1 . The date is not on the title-page nor at the 

] "Atquc utinam et alii cognoscere vellent nun ostentare out st quae nesciunt veluti 
fumum venditare non conarentur." 

* Campanus 1 translation- in Rat dolt's edition is. as follows: "Quanti rates quae dicuntur 
continuom habere proportionalilatem, sunt, quarum equ£ multiplicia apt equa sunt aut 
eque sibi sine interruptions addunt aut minuunt" (!), to which Campanus adds the note: 
" Com mil e proportion alia. sunt quorum omnia multiplicia equaliasunt continue proportional ia. 
Sed noluit ipsam diffinitionem proponete sub mac forma, quia tunc diffipiret idem per idem, 
apcrte (? a parte) tamen rei est istud cum sua difhnitione converiibile. " 

* "Euclidis Megarensis" Geometricorum Elementorum Libri XV. Campani Galli trans* 
alpini in eosdem commentariorum libri XV. Theonis Alexandrini Barthoiom&eo Zamberto 
Veneto inlerprete, in tredecim priores, commentationum libri XI 11. Hypsiclis Alexandria! in 
duos posteriores, eodem Barthoiomaco Zamberto Veneto interprete, commentariorum libri II." 
On the last page (.6t) is a similar statement of content, but wish the difference that the 
expression "ex Campani..,deinde Theon is... et Hypsiclis.../rWi'fra»ft&*/." For description 
see Weissentarn, p. 56 sqq- 



ioo INTRODUCTION [ch. vih 

end, but the letter of dedication to Francois Brieonnet by Jacques 
Leftvre is dated the day after the Epiphany, 15 16. The figures are 
in the margin. The arrangement of the propositions is as follows : 
first the enunciation with the heading Eudides ex Catnpano, then the 
proof with the note Campanus, and after that, as Campani additio, any 
passage found in the edition of Campanus' translation but not in the 
Greek text ; then follows the text of the enunciation translated from 
the Greek with the heading Euclides ex Zamberto, and lastly the proof 
headed Theo ex Zamberto. There are separate figures for the two proofs, 
This edition was reissued with few changes in 1537 and 1546 at Basel 
(apud Jehannem Heruagium), but with the addition of the Pkaenomena, 
Optica, Catoptrica etc. For the edition of 1537 the Paris edition of 
15 16 was collated with "a Greek copy" (as the preface says) by 
Christian Herlin, professor of mathematical studies at Strasshurg, 
who however seems to have done no more than correct one or two 
passages by the. help of the Basel editio princeps (1533), and add the 
Greek word in cases where Zamberti's translation of it seemed unsuit- 
able or inaccurate 
We now come to 

II. Editions of the Greek text, 

1 533 is the date of the editio princeps, the title-page of which reads 
as follows: 

ETKAEIAOT STOlXEIflN BIBA>- IE> 

EK TON 6EON02 ZYNOYSICN. 

Et? tov avrov to wp&TOV, ifyyriftdT&v UpoicXov /9*/9\. 5. 

Adiecta praefatiuncula in qua de disciplinis 

Mathematicis nonnihil. 

BASILEAE APVD 10 AN. HERVAGIVM ANNO 

M.D.XXXIII. MENSE SEPTEMBRI. 

The editor was Simon Grynaeus the elder (d. 1541), who, after 
working at Vienna and Ofen, Heidelberg and Tubingen, taught last 
of all at Basel, where theology was his main subject. His "prae- 
fatiuncula " is addressed to an Englishman, Cuthbert Tonstall { 1474- 
■559)i who, having studied first at Oxford, then at Cambridge, where 
he became Doctor of Laws, and afterwards at Padua, where in addi- 
tion he leamt mathematics — mostly from the works of Regiomontanus 
and Paciuolo— wrote a book on arithmetic 1 as "a farewell to the 
sciences," and then, entering politics, became Bishop of London and 
member of the Privy Council, and afterwards (1530) Bishop of Durham. 
Grynaeus tells us that he used two MSS. of the text of the Elements, 
entrusted to friends of his, one at Venice by "Lazarus Bayfius" 
(Lazare de BaJf, then the ambassador of the King of France at Venice), 
the other at Paris by " loann. Rvellius " (J ean Ruel, a French doctor 
and a Greek scholar), while the commentaries of Proclus were put at 

1 Dt arte supputandi libri quatuar. 



ch. viu] TRANSLATIONS AND EDITIONS 101 

the disposal of Grynaeus himself by " loann. Claymundus" at Oxford, 
Heiberg has been able to identify the two mss. used for the text ; 
they are (i) cod. Venetus Marcianus 301 and (2) cod. Paris, gr. 2343 
of the 16th c, containing Books 1. — xv., with some scholia which are 
embodied in the text. When Grynaeus notes in the margin the 
readings from " the other copy," this " other copy " is as a rule the 
Paris MS., though sometimes the reading of the Paris MS. is taken 
into the text and the " other copy " of the margin is the Venice MS. 
Besides these two mss. Grynaeus consulted Zamberti, as is shown by 
a number of marginal notes referring to " Zampertus " or to " latin um 
exemplar" in certain propositions of Books IX. — XL When it is con- 
sidered that the two MSS. used by Grynaeus are among the worst, it 
is obvious how entirely unauthoritative is the text of the editio princeps. 
Yet it remained the source and foundation of later editions of the 
Greek text for a long period, the editions which followed being 
designed, not for the purpose of giving, from other MSS., a text more 
nearly representing what Euclid himself wrote, but of supplying a 
handy compendium to students at a moderate price. 

1536. Orontius Finaeus (Oronce Fine) published at Paris {apurf 
Simonem Colinaeum) "demonstrations on the first six books of Euclid's 
elements of geometry," " in which the Greek text of Euclid himself is 
inserted in its proper places, with the Latin translation of Barth. 
Zamberti of Venice," which seems to imply that only the enunciations 
were given in Greek. The preface, from which Kastner quotes', says 
that the University of Paris at that time required, from all who 
aspired to the laurels of philosophy, a most solemn oath that they 
had attended lectures on the said first six Books. Other editions of 
Fine's work followed in 1544 and 1551. 

1545. The enunciations of the fifteen Books were published in 
Greek, with an Italian translation by Angelo Caiani, at Rome (apud 
Antonium Bladum Asulanum). The translator claims to have cor- 
rected the books and " purged them of six hundred things which did 
not seem to savour of the almost divine genius and the perspicuity of 
Euclid'" 

1549. Joachim Camerarius published the enunciations of the first 
six Books in Greek and Latin (Leipzig). The book, with preface, 
purports to be brought out by Rhaeticus (1514-1576), a pupil of 
Copernicus. Another edition with proofs of the propositions of the 
first three Books was published by Moritz Steinmetz in 1 577 (Leipzig) ; 
a note by the printer attributes the preface to Camerarius himself. 

155a loan. Scheubel published at Basel (also per loan. Her- 
vagium) the first six Books in Greek and Latin "together with true 
and appropriate proofs of the propositions, without the use of letters " 
(i.e. letters denoting points in the figures), the various straight lines 
and angles being described in words 1 . 

1557 (also 1558). Stephanus Gracilis published another edition 
(repeated 1 573, 1 578, 1 598) of the enunciations (alone) of Books I, — XV. 

1 Kastner, I. p. j6o. * Heiberg, vol. v. p. crii. ' Kastner, 1. p. 359. 



10a INTRODUCTION [ch. vm 

in Greek and Latin at Paris {apud Gulielmurn Cavellai). He remarks 
in the preface that for want of time he had changed scarcely anything 
in Books I. — VI., but ; n the remaining Books he had emended what 
seemed obscure or inelegant in the Latin translation, while he had 
adopted in its entirety the translation of Book x. by Pierre Mondore 
(PetrusMontaureus), published separately at Paris in 1551. Gracilis 
also added a few " scholia." 

1564. In this year Conrad Dasypodius (Rauchfuss), the inventor 
and maker of the clock in Strassburg cathedral, similar to the present 
one, which did duty from 1571 to 1789, edited (Strassburg, Chr. 
Mylius) (1) Book 1. of the Elements in Greek and Latin with scholia, 
(2) Book 11. in Greek and Latin with Barlaam's arithmetical version 
of Book II., and (3) the enunciations of the remaining Books III. — XIII. 
Book 1. was reissued with " vocabula quaedam geometrica " of Heron, 
the enunciations of all the Books of the Elements, and the other works 
of Euclid, all in Greek and Latin. In the preface to (1) he says that it 
had been for twenty-six years the rule of his school that all who were 
promoted from the classes to public lectures should learn the first 
Book, and that he brought it out, because there were then no longer 
any copies to be had, and in order to prevent a good and fruitful 
regulation of his school from falling through. In the preface to the 
edition of 157 1 he says that the first Book was generally taught in all 
gymnasia and that it was prescribed in his school for the first class. 
In the preface to (3) he tells us that he published the enunciations of 
Books in.— xill. in order not to leave his work unfinished, but that, as 
it would be irksome to carry about the whole work of Euclid in 
extenso, he thought it would be more convenient to students of 
geometry to learn the Elements if they were compressed into a smaller 
book. 

1620. Henry Briggs {of Briggs' logarithms) published the first 
six Books in Greek with a Latin translation after Commandinus, 
"corrected in many places" (London, G. Jones). 

1703 is the date of the Oxford edition by David Gregory which, 
until the issue of Heiberg and Menge's edition, was still the only 
edition of the complete works of Euclid 1 . In the Latin translation 
attached to the Greek text Gregory says that he followed Comman- 
dinus in the main, but corrected numberless passages in it by means 
of the books in the Bodleian Library which belonged to Edward 
Bernard (1638- 1 696), formerly Savilian Professor of Astronomy, who 
had conceived the plan of publishing the complete works of the ancient 
mathematicians in fourteen volumes, of which the first was to contain 
Euclid's Elements I. — xv. As regards the Greek text, Gregory tells us 
that he consulted, as far as was necessary, not a few MSS, of the better 
sort, bequeathed by the great Savile to the University, as well as the 
corrections made by Savile in his own hand in the margin of the Basel 
edition. He had the help of John Hudson, Bodley's Librarian, who 

1 ETKAEIAOT TA SOZOMBNA. Euclidis qnae supersunl omnia. Ex recensions 
Davidis Gregorii M.D. Asttonomiae Professoris Saviliani el R.S.S. Oxoniae, o Theatio 
Sheldoniano, An. Dora, mdccici. 



cm. vin] TRANSLATIONS AND EDITIONS 103 

punctuated the Basel text before it went to the printer, compared the 
Latin version with the Greek throughout, especially in the Elements 
and Data, and, where they differed or inhere he suspected the Greek text, 
consulted the Greek msS. and put their readings in the margin if 
they agreed with the Latin and, if they did not agree, affixed an 
asterisk in order that Gregory might judge which reading was geo- 
metrically preferable. Hence it is clear that no Greek MS., but the 
Basel edition, was the foundation of Gregory's text, and that Greek 
MSS. were only referred to in the special passages to which Hudson 
called attention. 

1 is 14-1 818. A most important step towards a good Greek text 
was taken by F. Peyrard, who published at Paris, between these years, 
in three volumes, the Elements and Data in Greek, Latin and French 1 . 
At the time (1808) when Napoleon was having valuable MSS. selected 
from Italian libraries and sent to Paris, Peyrard managed to get two 
ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 
304 was a ' so at Paris at the time, but all three were restored to their 
owners in 1 8 14). Peyrard noticed the excellence of Cod. Vat. 190, 
adopted many of its readings, and gave in an appendix a conspectus 
of these readings and those of Gregory's edition ; he also noted here 
and there readings from Vat. 1038 and various Paris MSS. He there- 
fore pointed the way towards a better text, but committed the error 
of correcting the Basel text instead of rejecting it altogether and 
starting afresh. 

1824-1825. A most valuable edition of Books I. — VI. is that of 
J. G. Camerer (and C. F, Hauber) in two volumes published at 
Berlin*. The Greek text is based on Peyrard, although the Basel 
and Oxford editions were also used. There is a Latin translation 
and a collection of notes far more complete than any other I have 
seen and well nigh inexhaustible. There is no editor or commentator 
of any mark who is not quoted from ; to show the variety of important 
authorities drawn upon by Camerer, I need only mention the following 
names : Proclus, Pappus, Tartaglia, Command inus, Clavius, Peletier, 
Barrow, Borelli, Watlis, Tacquet, Austin, Simson, Playfair. No words 
of praise would be too warm for this veritable encyclopaedia of 
information. 

1825. J. G. C. Neide edited, from Peyrard, the text of Books 
I. — VI., XI. and XII. (Halts Saxoniae). 

1826-9. The last edition of the Greek text before Heiberg's is 
that of E. F. August, who followed the Vatican MS. more closely 
than Peyrard did, and consulted at all events the Viennese MS. 
Gr, 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains 
Books I. — XIII. 

1 Eudidis quae supersuni. Las tEuvres a*Eudide t m Gree, en Latin ei en Fronfais 
d'aprls an msnuscrit tris-atvitn, qui Itait rerte* iwonnu jusqn'a nos jeurt. Par F, Peyrard. 
Ouvrijje approuv* par I'lnttitut de Franc* (Paris, chei M. Patris). 

1 Euclidis elemettforum hbri lex prwrrs graice et Inline tommentarie e scriptis veierum ae 
rteentforum mathtmatxarum ei Pjltiaertri maxime illustraii [Berolnii, suEnptibus G. Rcimeri). 
Tom. I. 1814 ; torn. II. t8i{. 



io4 INTRODUCTION [ch. viii 

III. Latin versions or commentaries after 1533. 

1545. Petrus Ramus (Pierre de la Ramee, 151 5-1572) is credited 
with a translation of Euclid which appeared in 1545 and again in 
1549 at Paris 1 . Ramus, who was more rhetorician and logician than 
geometer, also published in his Scholae mathematical ( iSS9i Frankfurt; 
1569, Base!) what amounts to a series of lectures on Euclid's Elements, 
in which he criticises Euclid's arrangement of his propositions, the 
definitions, postulates and axioms, all from the point of view of logic. 

1557. Demonstrations to the geometrical Elements of Euclid, six 
Books, by Peletarius (Jacques Peletier). The second edition (1610) 
contained the same with the addition of the "Greek text of Euclid"; 
but only the enunciations of the propositions, as well as the defini- 
tions etc., are given in Greek (with a Latin translation), the rest is 
in Latin only. He has some acute observations, for instance about 
the "angle" of contact 

•559- Johannes Buteo, or Borrel (1492-1573), published in an 
appendix to his book De quadratura circuit some notes " on the errors 
of Campanus, Zambertus, Orontius, Peletarius, Pena, interpreters of 
Euclid." Buteo in these notes proved, by reasoned argument based 
on original authorities, that Euclid himself and not Theon was the 
author of the proofs of the propositions. 

1 566. Franciscus Flussates Candalla (Francois de Foix, Comte de 
Candale, 1 502-1 594) "restored" the fifteen Books, following, as he 
says, the terminology of Zamberti's translation from the Greek, but 
drawing, for his proofs, on both Campanus and Theon (i.e. Zamberti) 
except where mistakes in them made emendation necessary. Other 
editions followed in 1578, 1602, 1695 (in Dutch). 

1572. The most important Latin translation is that of Com- 
mand irtus (1509-1575) of Urbino, since it was the foundation of most 
translations which followed it up to the time of Peyrard, including 
that of Simson and therefore of those editions, numerous in England, 
which give Euclid "chiefly after the text of Simson." Simson 's first 
(Latin) edition (1756) has "ex versione Latina Federici Commandini" 
on the title-page. Commandinus not only followed the original Greek 
more closely than his predecessors but added to his translation some 
ancient scholia as well as good notes of his own. The title of his 
work is 

Euclidis elementorum libri xv, una cum sckeliis antiquis. 
A Federico Contmandino Urbinate nuper in latinum conversi, 
commentariisque quibusdam illustrati (Pisauri, apud Camillum 
Francischinum). 

He remarks in his preface that Orontius Finaeus had only edited 
six Books without reference to any Greek ms., that Peletarius had 
followed Campanus' version from the Arabic rather than the Greek 
text, and that Candalla had diverged too far from Euclid, having 
rejected as inelegant the proofs given in the Greek text and 
substituted faulty proofs of his own. Commandinus appears to have 

1 Described by Boncompagni, BuiUitino, it. p. 3S9. 



ch. vm] TRANSLATIONS AND EDITIONS 105 

used, in addition to the Basel editio princeps, some Greek ms., so far 
not identified ; he also extracted his " scholia antiqua " from a US. 
of the class of Vat. 192 containing the scholia distinguished by 
Heiberg as " Schol. Vat." New editions of Commandinus' translation 
followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and 
Stone), 1756 (Books 1. — VI., XI., XII. in Latin and English, by Simson), 
1763 (Keill). Besides these there were many editions of parts of the 
whole work, e.g. the first six Books. 

[574. The first edition of the Latin version by Clavius' 
(Christoph Klau [?], born at Bamberg 1537, died 1612) appeared 
in 1574, and new editions of it in 1589, 1591, 1603, 1607, 1612. It is 
not a translation, as Clavius himself states in the preface, but it 
contains a vast amount of notes collected from previous commentators 
and editors, as well as some good criticisms and elucidations of his 
own. Among other things, Clavius finally disposed of the error by 
which Euclid had been identified with Euclid of Megara. He speaks 
of the differences between Campanus who followed the Arabic 
tradition and the " commentaries of Theon," by which he appears to 
mean the Euclidean proofs as handed down by Theon ; he complains 
of predecessors who have either only given the first six Books, or 
have rejected the ancient proofs and substituted worse proofs of their 
own, but makes an exception as regards Commandinus, " a geometer 
not of the common sort, who has lately restored Euclid, in a Latin 
translation, to his original brilliancy." Clavius, as already stated, did 
not give a translation of the Elements but rewrote the proofs, com- 
pressing them or adding to them, where he thought that he could 
make them clearer. Altogether his book is a most useful work. 

1 62 1. Henry Savile's lectures {Praelectiones tresdecttn in prin- 
cipium EUtnentorum Etulidis Oxoniae habitae MDC.XX., Oxonii 1621), 
though they do not extend beyond I. 8, are valuable because they 
grapple with the difficulties connected with the preliminary matter, 
the definitions etc., and the tacit assumptions contained in the first 
propositions, 

1654 Andre 1 Tacquet's Elementa geometriae planae et solidae 
containing apparently the eight geometrical Books arranged for 
general use in schools. It came out in a large number of editions up 
to the end of the eighteenth century. 

1655, Barrow's Euclidis Eiementorum Libri XV breviter demon- 
strati is a book of the same kind. In the preface (to the edition of 
[659) he says that he would not have written it but for the fact that 
Tacquet gave only eight Books of Euclid. He compressed the work 
into a very small compass (less than 400 small pages, in the edition 
of 1659, for the whole of the fifteen Books and the Data) by abbre- 
viating the proofs and using a large quantity of symbols (which, he 
says, are generally Oughtred's). There were several editions up to 
1732 (those of 1660 and 1732 and one or two others are in English). 

1 EudiMs clem/ntor um libri XV. Aictuit xvi. de selidorum regaiarium comparatism. 
Omnes pefipituis dtmonstratiimibm, accurattsqut nhotiii iltustrati. Auclorc ChristophoTO 
Ctaaio (Romae, apud Vinccotium Accoltum), 1 vols. 



to6 INTRODUCTION [ch. vm 

1658. Giovanni Alfonso Borelli (1608- 1 679) published Euclides 
restitutus, on apparently similar lines, which went through three more 
editions (one in Italian, 1663). 

166a Claude Francois Milliet Dechales' eight geometrical Books 
of Euclid's Elements made easy. Dechales' versions of the Elements 
had great vogue, appearing in French, Italian and English as well 
as Latin. Riccardi enumerates over twenty editions. 

1733. Saccheri's Euclides ab omni naevo vindicatus sive const us 
geometrkus quo stabiliuntur prima ipsa geomelriae principia is 
important for his elaborate attempt to grove the parallel-postulate, 
forming an important stage in the history of the development of non- 
Euclidean geometry. 

1756. Simson's first edition, in Latin and in English. The Latin 
title is 

Euclidis elementorum libri priores sex, item undecimus et duo- 
decimus, ex versione latina Federici Commandtni; sublatis iis 
quibus olim libri hi a Tkeone, aliisve, vitiali sunt, et quibusdam 
Euclidis demonstrationibus restitutis. A Roberto Simson M.D. 
Glasguae, in aedibus Academicis excudebant Robertus et Andreas 
Foulis, Academiae typographi. 

1802. Euclidis elementorum libri priores XII ex Commandini et 
Gregorii versianibus latinis. In usum juventutis Academicae..,hy 
Samuel Horsley, Bishop of Rochester. (Oxford, Clarendon Press.) 

IV. Italian versions or commentaries. 

1543. Tartaglia's version, a second edition of which was pub- 
lished in 1565', and a third in 1585. It does not appear that he used 
any Greek text, for in the edition of 1565 he mentions as available 
only "the first translation by Campano," "the second made by 
Bartolomeo Zamberto Veneto who is still alive," "the editions of 
Paris or Germany in which they have included both the aforesaid 
translations," and "our own translation into the vulgar (tongue)." 

IS7S- Commandinus' translation turned into Italian and revised 
by him. 

1613. The first six Books "reduced to practice" by Pietro 
Antonio Cataldi, re-issued in 1620, and followed by Books VH, — ix. 
(1621) and Book X. (1625). 

1663. Borelli's Latin translation turned into Italian by Domenico 
Magni. 

1680. Euclide restitute by Vitale Giordano. 

169a Vincenzo Vivian i's Eletnenti piani e solid t di Euclide 
(Book v. in 1674). 

1 The title-page of the edition of 1 56; is ss follows : Euclide Megarcnse phihsophv. sail 
introduttorc deile identic msthematicc, diligtnttmente rasiettatc, et alia inicgrita ridotls, per it 
degno pr&fessore di tat identic Nicola Tartatca Briscians, scconde U due tratioltioni. con una 
ampla espositione delle istcsso tradottore di nuouo aggiutita. ialmente ckiara, cat ogni mediocre 
ingigno, stitta la notiiia, aver tvffragio di alcutT a/tra icientia con facUita jrrd capacc a 
peter la inttnderi. In Venetia, Appresso Curtio Troiano, 1565. 



ch. vm] TRANSLATIONS AND EDITIONS 107 

173 1. Elementi geomttrici ptant e solid 'i di Eu elide by Guido 
Grandi. No translation, but an abbreviated version, of which new 
editions followed one another up to 1806. 

1749. Italian translation of" Dechales with Ozanam's corrections 
and additions, re-issued 1785, 1797. 

1752. Leonardo Ximenes (the first six Books). Fifth edition, 
1 819. 

181 8. Vincenzo Flauti's Corso di geometries eUtnentare e sublime 
(4 vols.) contains (Vol. I.) the first six Books, with additions and a 
dissertation on Postulate 5, and (Vol. II.) Books xi„ XH. Flauti 
also published the first six Books in 1827 and the Elements of geometry 
of Euclid m 1843 and 1854 

V. German. 

1558. The arithmetical Books vir. — ix. by Scheubel" (cf. the 
edition of the first six Books, with enunciations in Greek and Latin, 
mentioned above, under date 1 5 50). 

1562. The version of the first six Books by Wilhelm Holtzmann 
(Xylander)*. This work has its interest as the first edition in German, 
but otherwise it is not of importance. Xylander tells us that it was 
written for practical people such as artists, goldsmiths, builders etc., 
and that, as the simple amateur is of course content to know facts, 
without knowing how to prove them, he has often left out the proofs 
altogether. He has indeed taken the greatest possible liberties with 
Euclid, and has not grappled with any of the theoretical difficulties, 
such as that of the theory of parallels. 

1651. Heinrich Hoffmann's Teutscher Euclides (2nd edition 1653), 
not a translation. 

1694. Ant Ernst Burkh. v. Pirckenstein's Teutsch Redender 
Euclides (eight geometrical Books), "for generals, engineers etc." 
"proved in a new and quite easy manner." Other editions 1699, 
1744. 

1697. Samuel Reyher's In teutscher Sfiraclie vorgestellter Euclides 
(six Books), "made easy, with symbols algebraical or derived from the 
newest art of solution." 

1714. Euclidu xv Bilcker teutsch, "treated in a special and 
brief manner, yet completely," by Chr. Schessler (another edition in 
1729). 

1773. The first six Books translated from the Greek for the 
use of schools by J. F, Lorenz. The first attempt to reproduce 
Euclid in German word for word. 

1 78 1. Books XI., xji. by Lorenz (supplementary to the pre- 
ceding). Also Euklid's Etemente fUnfzekn Biicker translated from 

1 Das libcndochl und taunt buck da kochherumbun Mathematici Euclidis Mtgartnsis... 
durck Magistrum Jahaun Sckeybf, der loblichen univcrsittt zu Tubingen, des EucHdis und 
Arithmetic Ordinarien t uuss dem lutein ins teutsch gebra£kt 

* Di* seeks erste Hucher Euclidis vom an fang oder grund der Geometry. ..Asess Grserhtscker 
sfiraek in die Teutsch gebraekt aigentlick erkldrt. . . Demassen vcr/uals in Teutscher spraek nie 
imhen warden... Durch Wilktlm Holttman gtnattt Xylander von Augsfurg. Getnickht m 



io8 INTRODUCTION [ch. viii 

the Greek by Lorenz (second edition 1798; editions of 1809, 1818, 
1824 by Mollweide, of 1840 by Dippe). The edition of 1824, and 
I presume those before it, are shortened by the use of symbols and 
the compression of the enunciation and "setting-out" into one. 

1807. Books 1.— vi., xi. ( xil. "newly translated from the Greek," 
by J. K. F. Hauff. 

1828. The same Books by Joh. Jos. Ign. Hoffmann "as guide 
to instruction in elementary geometry," followed in 1832 by observa- 
tions on the text by the same editor. 

1833. Die Geometric des Euklid und das Wesen derselben by 
E. S. Unger; also 1838, 1851. 

1901. Max Simon, Euclid und die seeks planimetrischen Bucket. 

VI, French. 

1564-1566. Nine Books translated by Pierre Forcadel, a pupil 
and friend of P. de la Ramee. 

1604. The first nine Books translated and annotated by Jean 
Errard de Bar-Ie-Duc ; second edition, 1605. 

161 5. Denis Hen r ion's translation of the 15 Books (seven 
editions up to 1676), 

1639. The first six Books "demonstrated by symbols, by a 
method very brief and intelligible," by Pierre Hengone, mentioned 
by Barrow as the only editor who, before him, had used symbols for 
the exposition of Euclid. 

1672. Eight Books "rend us plus faciles" by Claude Francois 
Mi Diet Dec hales, who also brought out Les ilemens d'Euclide ex- 
pliquis d'une maniere nouvelie et trh facile, which appeared in many 
editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), 
and was translated into Italian (1749 etc.) and English (by William 
Halifax, 1685). 

1 804. In this year, and therefore before his edition of the Greek 
text, F. Peyrard published the Elements literally translated into 
French. A second edition appeared in 1 809 with the addition of the 
fifth Book, As this second edition contains Books 1. — vi. XL, xn. 
and x. 1, it would appear that the first edition contained Books 1. — iv., 
VI., XI., XII. Peyrard used for this translation the Oxford Greek text 
and Simson. 

VII. Dutch. 

1606. Jan Pieterszoon Dou (six Books). There were many later 
editions. Kastner, in mentioning one of 1702, says that Dou explains 
in his preface that he used Xylander's translation, but, having after- 
wards obtained the French translation of the six Books by Errard 
de Bar-le-Duc (see above),' the proofs in which sometimes pleased 
him more than those of the German edition, he made his Dutch 
version by the help of both. 

1 61 7. Frans van Schooten, "The Propositions of the Books of 
Euclid's Elements"; the fifteen Books in this version " enlarged " by 
Jakob van Leest in 1662. 

1695. C. J. Vooght, fifteen Books complete, with Candalla's " 16th." 



ch. vin] TRANSLATIONS AND EDITIONS 109 

1702. Kendfik Coets, six Books (also in Latin, 1692); several 
editions up to 1752. Apparently not a translation: but an edition for 
school use. 

1763. Pybo Steenstra, Books I. — vr„ XI., XII., likewise an abbre- 
viated version, several times reissued until 1825. 

VIII. English. 
1570 saw the first and the most important translation, that of Sir 
Henry Billingsley. The title-page is as follows : 

THE ELEMENTS 

OF GEOMETRIE 

of the most auncient Philosopher 

E VCLWE 

of Megara 

Faithfully (now first) translated into the English toung, 

by H. Billingsley, Citizen of London. Whereunto are annexed 

certaine Scholies, Annotations, and Inuentions, of the best 

Mathematiciens, both of time past, and in this our age. 

With a very fruitfull Preface by M. I. Dee, specifying the 
chiefe Mathetnaticall Sciiees, what they are, and whereunto 
commodious: where, also, are disclosed certaine new Secrets 
Mathetnaticall and Afechanicall, vntill these our dales, greatly 
missed. 

Imprinted at London by John Daye. 

The Preface by the translator, after a sentence observing that with- 
out the diligent study of Euclides Efementes it is impossible to attain 
unto the perfect knowledge of Geometry, proceeds thus. " Wherefore 
considering the want and lacke of such good authors hitherto in our 
Englishe tounge, lamenting also the negligence, and lacke of zeale to 
their countrey in those of our nation, to whom God hath geuen both 
knowledge and also abilitie to translate into our tounge, and to 
publishe abroad such good authors and bookes (the chiefe instrumentes 
of all learninges): seing moreouer that many good wittes both of 
gentlemen and of others of all degrees, much desirous and studious of 
these artes, and seeking for them as much as they can, sparing no 
paines, and yet frustrate of their intent, by no meanes attaining to 
that which they seeke : I haue for their sakes, with some charge and 
great trauaile, faithfully translated into our vulgar e touge, and set 
abroad in Print, this booke of Euclide. Whereunto I haue added 
easie and plaine declarations and examples by figures, of the defini- 
tions. In which booke also ye shall in due place finde manifolde 
additions, Scholies, Annotations, and Inuentions; which I haue 
gathered out of many of the most famous and chiefe Mathematicies, 
both of old time, and in our age : as by diligent reading it in course, 
ye shall well perceaue...," 

It is truly a monumental work, consisting of 464 leaves, and there- 
fore 928 pages, of folio size, excluding the lengthy preface by Dee. 
The notes certainly include all the most important that had ever been 



tto INTRODUCTION [cm. vin 

written, from those of the Greek commentators, Proclus and the others 
whom he quotes, down to those of Dee himself on the la3t books. 
Besides the fifteen Books, Billingsley included the "sixteenth" added 
by Candalla, The print and appearance of the book are worthy of its 
contents ; and, in order that it may be understood how no pains were 
spared to represent everything in the clearest and most perfect form, 
I need only mention that the figures of the propositions in Book XI. 
are nearly all duplicated, one being the figure of Euclid, the other an 
arrangement of pieces of paper (triangular, rectangular etc.) pasted at 
the edges on to the page of the book so that the pieces can be turned 
up and made to show the real form of the solid figures represented. 

Billingsley was admitted Lady Margaret Scholar of St John's 
College, Cambridge, in 1551, and he is also said to have studied at 
Oxford, but he did not take a degree at either University. He was 
afterwards apprenticed to a London haberdasher and rapidly became 
a wealthy merchant. Sheriff of London in 1584, he was elected Lord 
Mayor on 31st December, 1596, on the death, during his year of office, 
of Sir Thomas Skinner. From 1589 he was one of the Queen's four 
" customers," or farmers of customs, of the port of London. In 1 591 
he founded three scholarships at St John's College for poor students, 
and gave to the College for their maintenance two messuages and 
tenements in Tower Street and in Mark Lane, Allhallows, Barking. 
He died in 1606. 

1651. Elements of Geometry. The first VI Bootks: In a compen- 
dious form contracted and demonstrated by Captain Thomas Rudd, with 
the mathematicall preface of John Dee (London). 

1660. The first English edition of Barrow's Euclid (published in 
Latin in 1655)1 appeared in London. It contained "the whole fifteen 
books compendiously demonstrated"; several editions followed, in 
1705, 1722, 1732, 1751. 

1 66 1. Euclid 's Elements of Geometry, with a supplement of divers 
Propositions and Corollaries. To which is added a Treatise of regular 
Soli lis by Campane and Ftussat ; likewise Euclid's Data and Marinus 
his Preface. Also a Treatise of the Divisions of Superficies, ascribed to 
Machomet Bagdedine, but published by Commandine at the request of 
J, Dee of London, Published by care and industry of John Leeke and 
Geo. Serle, students in the Math. (London). According to Potts this 
was a second edition of Billingsley's translation. 

1685. William Halifax's version of Dechales' " Elements of Euclid 
explained in a new but most easy method " (London and Oxford). 

1705. The English Euclide; being tlie first six Elements of 
Geometry, translated out of the Greek, with annotations and useful/ 
supplements by Edmund Scarburgh (Ox ford ). A noteworthy and 
useful edition. 

1708. Books I. — VL, XL, xii., translated from Command in us' Latin 
version by Dr John Keill, Savilian Professor of Astronomy at Oxford. 

Keill complains in his preface of the omissions by such editors as 
Tacquet and Dechales of many necessary propositions (e.g. VI. 27 — 29), 
and of their substitution of proofs of their own for Euclid's. He praises 
Barrow's version on the whole, though objecting to the " algebraical " 



ca. viii] TRANSLATIONS AND EDITIONS in 

form of proof adopted in Book n., and to the excessive use of notes 
and symbols, which (he considers) make the proofs too short and 
thereby obscure: his edition was therefore intended to hit a proper 
mean between Barrow's excessive brevity and Clavius' prolixity. 

Keill's translation was revised by Samuel Cunn and several times 
reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 
the twelfth. 

1714. W. Whiston's English version (abridged) of The Elements 
of Euclid with select theorems out of Archimedes by the learned Andr. 
Tacquet. 

1756. Simson's first English edition appeared in the same year as 
his Latin version under the title : 

The Elements of Euclid, vis. the first six Books together with 
the eleventh and twelfth. In this Edition the Errors by which 
Theon or others have long ago vitiated these Books are corrected and 
some of Euclid's Demonstrations are restored. By Robert Simson 
(Glasgow). 

As above stated, the Latin edition, by its title, purports to be " ex 
version© latina Federici Commandini," but to the Latin edition, as well 
as to the English editions, are appended 

Notes Critical and Geometrical ; containing an Account of those 
things in which this Edition differs from the Greek text; and the 
Reasons of the Alterations which have been made. As also Obser- 
vations on some of the Propositions. 

Simson says in the Preface to some editions (e.g. the tenth, of 
1799) that "the translation is much amended by the friendly assistance 
of a learned gentleman." 

Simson's version and his notes are so well known as not to need 
any further description. The book went through some thirty suc- 
cessive editions. The first five appear to have been dated 1756, 1762, 
1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, 
the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844. 
The Data "in like manner corrected '' was added for the first time in 
the edition of 1762 (the first octavo edition). 

1781, 1788. In these years respectively appeared the two volumes 
containing the complete translation of the whole thirteen Books by 
James Williamson, the last English translation which reproduced 
Euclid word for word. The title is 

The Elements of Euclid, with Dissertations intended to assist 
and encourage a critical examination of these Elements, as the most 
effectual means of establishing a j'uster taste upon mathematical 
subjects than that which at present prevails. By James Williamson. 
In the first volume (Oxford, 1781) he is described as " M.A. 
Fellow of Hertford College," and in the second (London, printed by 
T. Spilsbury, 1788) as "B.D." simply. Books v., VI. with the Con- 
clusion in the first volume are paged separately from the rest 

1 78 1 . 4 n examination of the first six Books of Euclid's Elements, 
by William Austin (London). 

'79S- John Playfair's first edition, containing "the first six Books 
of Euclid with two Books on the Geometry of Solids." The book 



ii3 INTRODUCTION [ch. vm 

reached a fifth edition in 1819, an eighth in 1831, a ninth in 1836, and 
a tenth in 1846. 

1826. Riccardi notes under this date Euclid's Elements of Geo- 
metry containing the whole twelve Books translated into English, from the 
edition of Peyrard, by George Phillips. The editor, who was President 
of Queens' College, Cambridge, 1857-1892, was born in 1804 and 
matriculated at Queens' in 1 826, so that he must have published the 
book as an undergraduate. 

1828. A very valuable edition of the first six Books is that of 
Dionysius Lardner, with commentary and geometrical exercises, to 
which he added, in place of Books XI., XII., a Treatise on Solid 
Geometry mostly based on Legend re, Lardner compresses the pro- 
positions by combining the enunciation and the setting-out, and he 
gives a vast number of riders and additional propositions in smaller 
print The book had reached a ninth edition by 1846, and an eleventh 
by 1855. Among other things, Lardner gives an Appendix "on the 
theory of parallel lines," in which he gives a short history of the 
attempts to get over the difficulty of the parallel' postulate, down to 
that of Legendre. 

1833. T. Perronet Thompson's Geometry without axioms, or the 
first Book of Euclid's Elements with alterations and notes ; and an 
intercalary book in which the straight line and plane are derived from 
properties of the sphere, with an appendix containing notices of methods 
proposed for getting over the difficulty in the twelfth axiom of Euclid. 

Thompson (1783-1869) was 7th wrangler 1802, midshipman 1803, 
Fellow of Queens* College, Cambridge, 1804, and afterwards general 
and politician. The book went through several editions, but, having 
been well translated into French by Van Tenac, is said to have 
received more recognition in France than at home. 

1 845. Robert Potts' first edition (and one of the best) entitled : 

Euclid's Elements of Geometry chiefly from the text of 
Dr Simson with explanatory notes... to which is prefixed an 
introduction containing a brief outline of the History of Geometry. 
Designed for the use of the higher forms in Public Schools and 
students in the Universities (Cambridge University Press, and 
London, John W. Parker), to which was added (1847) An 
Appendix to the larger edition of Euclid's Elements of Geometry, 
containing additional notes on the Elements, a short tract on trans- 
versals, and hints for the solution of the problems etc. 
1862. Todhunter's edition. 

The later English editions 1 will not attempt to enumerate; their 
name is legion and their object mostly that of adapting Euclid for school 
use, with all possible gradations of departure from his text and order. 

IX, Spanish. 
1576. The first six Books translated into Spanish by Rodrigo 
Camorano. 

1637. The first six Books translated, with notes, by L. Carduchi. 

1689. Books 1.— vi, XI, XII, translated and explained by Jacob 
Knesa. 



ch. vni] TRANSLATIONS AND EDITIONS 113 

X. Russian. 

1739. Ivan Astaroff* (translation from Latin). 

1 789. Pr. Suvoroff and Yos. Nikitin {translation from Greek). 

1880. Vachtchenko-Zakhartchenko. 

{181 7, A translation into Polish by Jo. Czecha.) 

XI. Swedish. 

1744. Mitten Stromer, the first six Books ; second edition 1748. 
The third edition (1753) contained Books XI.— XU. as well; new 
editions continued to appear till 1884, 

1836. H. Falk, the first six Books. 

1844, 1845, 1859. P. R. Brakenhjelm, Books I. — VI., XL, xn. 

1850. F. A. A. Lundgren. 

1850. H. A. Witt and M. E. Areskong, Books I.— VI., XI., XII. 

XII. Danish. 

1745. Ernest Gottlieb Ziegenbalg. 
1803. H. C. Linderup, Books I. — VI. 

XIII. Modern Greek. 
1820. Benjamin of Lesbos, 

I should add a reference to certain editions which have appeared 
in recent years. 

A Danish translation (Euklid's Eletnenter oversat af Thyra Eibe) 
was completed in 1912 ; Books I. — II. were published (with an Intro- 
duction by Zeuthen) in 1897, Books in. — IV. in 1900, Books v. — VI. 
in 1904, Books VII. — XIII. in 19 1 2, 

The Italians, whose great services to elementary geometry are 
more than once emphasised in this work, have lately shown a note- 
worthy disposition to make the ipsissima verba of Euclid once more 
the object of study. Giovanni Vacca has edited the text of Book I. 
(// prima libro degli Elementi. Testo greco, versione italiana, intro- 
duzione e note, Firenze 1 916,) Federigo Enriques has begun the 
publication of a complete Italian translation (Gli Elementi d' Enclide 
e la critica antica e moderna); Books I. — IV. appeared in 1925 (Alberto 
Stock, Roma). 

An edition of Book I. by the present writer was published in 1918 
{Euclid in Greek \ Book [., with Introduction and Notes, Camb. Univ. 
Press). 









CHAPTER IX. 

j I, ON THE NATURE OF ELEMENTS. 

It would not be easy to find a more lucid explanation of the terms 
element and elementary, and of the distinction between them, than 
is found in Prod us \ who is doubtless, here as so often, quoting 
from Geminus. There are, says Proclus, in the whole of geometry 
certain leading theorems, bearing to those which follow the relation of 
a principle, all- pervading, and furnishing proofs of many properties. 
Such theorems are called by the name of elements ; and their function 
may be compared to that of the letters of the alphabet in relation to 
language, letters being indeed called by the same name in Greek 
(orotyeta). 

The term elementary, on the other hand, has a wider application : 
it is applicable to things " which extend to greater multiplicity, and, 
though possessing simplicity and elegance, have no longer the same 
dignity as the elements, because their investigation is not of general 
use in the whole of the science, e.g. the proposition that in triangles 
the perpendiculars from the angles to the transverse sides meet in a 
point." 

" Again, the term element is used in two senses, as Menaechmus 
says. For that which is the means of obtaining is an element of that 
which is obtained, as the first proposition in Euclid is of the second, 
and the fourth of the fifth. In this sense many things may even be 
said to be elements of each other, for they are obtained from one 
another. Thus from the fact that the exterior angles of rectilineal 
figures are (together) equal to four right angles we deduce the number 
of right angles equal to the internal angles (taken together)*, and 
vice versa. Such an element is like a lemma. But the term element is 
otherwise used of that into which, being more simple, the composite is 
divided ; and in this sense we can no longer say that everything is an 
element of everything, but only that things which are more of the 
nature of principles are elements of those which stand to them in the 
relation of results, as postulates are elements of theorems. It is 

1 Proclus, Coium, on Eucl. I., ed. Friedlein, pp. Jssqq. 

* t4 TrkTjSin rw irrit dfi0a(j tnur. If the te*t is right, we must apparently take it as "the 
number of the angles equal to right angles that there are inside/' i.e. that are made up by 
the internal angles. 



ch. ix. J i] ON THE NATURE OF ELEMENTS 115 

according to this signification of the term element that the elements 
found in Euclid were compiled, being partly those of plane geometry, 
and partly those of stereometry. In like manner many writers have 
drawn up elementary treatises in arithmetic and astronomy. 

" Now it is difficult, in each science, both to select and arrange in 
due order the elements from which all the rest proceeds, and into 
which all the rest is resolved. And of those who have made the 
attempt some were able to put together more and some less ; some 
used shorter proofs, some extended their investigation to an indefinite 
length ; some avoided the method of reductio ad absurdum, some 
avoided proportion-, some contrived preliminary steps directed against 
those who reject the principles ; and, in a word, many different 
methods have been invented by various writers of elements. 

"It is essential that such a treatise should be rid of everything 
superfluous (for this is an obstacle to the acquisition of knowledge) ; 
it should select everything that embraces the subject and brings it to 
a point (for this is of supreme service to science) ; it must have great 
regard at once to clearness and conciseness (for their opposites trouble 
our understanding); it must aim at the embracing of theorems in 
general terms (for the piecemeal division of instruction into the more 
partial makes knowledge difficult to grasp). In all these ways 
Euclid's system of elements will be found to be superior to the rest ; 
for its utility avails towards the investigation of the primordial 
figures 1 , its clearness and organic perfection are secured by the 
progression from the more simple to the more complex and by the 
foundation of the investigation upon common notions, while generality 
of demonstration is secured by the progression through the theorems 
which are primary and of the nature of principles to the things sought. 
As for the things which seem to be wanting, they are partly to be 
discovered by the same methods, like the construction of the scalene 
and isosceles (triangle), partly alien to the character of a selection of 
elements as introducing hopeless and boundless complexity, like the 
subject of unordered irrationals which Apoilonius worked out at 
length", and partly developed from things handed down (in the 
elements) as causes, like the many species of angles and of lines. 
These things then have been omitted in Euclid, though they have 
received full discussion in other works ; but the knowledge of them is 
derived from the simple (elements)." 

Proclus, speaking apparently on his own behalf, in another place 
distinguishes two objects aimed at in Euclid's Elements. The first 
has reference to the matter of the investigation, and here, like a good 
Platonist, he takes the whole subject of geometry to be concerned 
with the "cosmic figures," the five regular solids, which in Book XHi. 

1 rwt Apxix&r <jyv!i&Twi\ by which Procius probably means the regular polyhedra 
(Tannery, p. l43ff.). 

■ We have no more than the most obscure indications of the character of this work in an 
Arabic MS. analysed by Wocpcke, Essai d*une rtstiiutitm dt travaux pirdus tfApoUojnus 
sur Us quantities irraiwntlUs d'aprls des indications tirhs tTun manuscrit arabe in Mtmvirtx 
prlttnttt e fatadMit da Mentis, xjv. 658—710, Paris, 1856. Cf. Cantor, Gtsth. d. Math. 
t 3 , pp. 34& — 9: details are also given in my notes to Book X. 



n6 INTRODUCTION [ch ex. § i 

are constructed, inscribed in a sphere and compared with one another. 
The second object is relative to the learner; and, from this standpoint, 
the elements may be described as "a means of perfecting the learner's 
understanding with reference to the whole of geometry. For, starting 
from these (elements), we shall be able to acquire knowledge of the 
other parts of this science as well, while without them it is impossible 
for us to get a grasp of so complex a subject, and knowledge of the 
rest is unattainable. As it is, the theorems which are most of the 
nature of principles, most simple, and most akin to the first hypotheses 
are here collected, in their appropriate order ; and the proofs of all 
other propositions use these theorems as thoroughly well known, and 
start from them. Thus Archimedes in the books on the sphere and 
cylinder, Apollonius, and all other geometers, clearly use the theorems 
proved in this very treatise as constituting admitted principles 1 ." 

Aristotle too speaks of elements of geometry in the same sense. 
Thus: "in geometry it is well to be thoroughly versed in the 
elements* " ; " in general the first of the elements are, given the 
definitions, e.g. of a straight line and of a circle, most easy to prove, 
although of course there are not many data that can be used to 
establish each of them because there are not many middle terms'"; 
"among geometrical propositions we call those 'elements' the proofs of 
which are contained in the proofs of all or most of such propositions'"; 
"(as in the case of bodies), so in like manner we speak of the elements 
of geometrical propositions and, generally, of demonstrations ; for the 
demonstrations which come first and are contained in a variety of 
other demonstrations are called elements of those demonstrations... 
the term element is applied by analogy to that which, being one and 
small, is useful for many purposes •." 



§ x ELEMENTS ANTERIOR TO EUCLID'S. 

The early part of the famous summary of P rod us was no doubt 
drawn, at least indirectly, from the history of geometry by Eudemus ; 
this is generally inferred from the remark, made just after the mention 
of PhiHppus of Medma, a disciple of Plato, that "those who have 
written histories bring the development of this science up to this 
point." We have therefore the best authority for the list of writers of 
elements given in the summary. Hippocrates of Chios (fl, in second 
half of 5th c.) is the first; then Leon, who also discovered diorismi, 
put together a more careful collection, the propositions proved in it 
being more numerous as well as more serviceable*. Leon was a little 
older than Eudoxus (about 408-355 n.C.) and a little younger than 
Plato (428/7-347/6 B.C.), but Hid not belong to the latter's school. The 

J Proclus, pp. 70, 10— 71, 31. 

5 Tapks vm. 14,163b 13.' % Topics vm. 3, 158 b 35. ' Mttaph. 908 » )j, 

' Mttaph. 1014 a 35 — b 5, 

Proclus, p. 66, 10 UMTre rbv Alojrra xal rd ffT«x*ia rvpfftiftu t^j re w\f)8ti teal r£ xpfi? 



ch. ix. (»] ELEMENTS ANTERIOR TO EUCLID'S 117 

geometrical text-book of the Academy was written by Theudius of 
Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of 
Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus 
consorted together in the Academy and carried on their investigations 
in common. Theudius " put together the elements admirably, making 
many partial (or limited) propositions more general 1 ." Eudemus 
mentions no text- book after that of Theudius, only adding that Her- 
motimus of Colophon "discovered many of the elements 1 ." Theudius 
then must be taken to be the immediate precursor of Euclid, and no 
doubt Euclid made full use of Theudius as well as of the discoveries of 
Hermotimus and all other available material. Naturally it is not in 
Euclid's Elements that we can find much light upon the state of the 
subject when he took it up ; but we have another source of informa- 
tion in Aristotle. Fortunately for the historian of mathematics, 
Aristotle was fond of mathematical illustrations ; he refers to a con- 
siderable number of geometrical propositions, definitions etc., in a 
way which shows that his pupils must have had at hand some text- 
book where they could find the things he mentions; and this text -book 
must have been that of Theudius. Heiberg has made a most valuable 
collection of mathematical extracts from Aristotle*, from which much 
is to be gathered as to the changes which Euclid made in the methods 
of his predecessors ; and these passages, as well as others not included 
in Heiberg's selection, will often be referred to in the sequel. 

§3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, 
AND AXIOMS. 

On no part of the subject does Aristotle give more valuable 
information than on that of the first principles as, doubtless, generally 
accepted at the time when he wrote. One long passage in the 
Posterior Analytics is particularly full and lucid, and is worth quoting 
m txtenso. After laying it down that every demonstrative science 
starts from necessary principles', he proceeds': 

" By first principles in each genus I mean those the truth of which 
it is not possible to prove. What is denoted by the first (terms) and 
those derived from them is assumed ; but, as regards their existence, 
this must be assumed for the principles but proved for the rest. Thus 
what a unit is, what the straight (line) is, or what a triangle is (must 
be assumed); and the existence of the unit and of magnitude must 
also be assumed, but the rest must be proved. Now of tie premisses 
used in demonstrative sciences some are peculiar to each science and 
others common (to all), the latter being common by analogy, for of 
course they are actually useful in so far as they are applied to the sub- 
ject-matter included under the particular science. Instances of first 

1 Proclus, p. 67, 14 «oi yip ri jtoix'" 1 *"\ut avriragtr ml ro\Wl rur ntpiKwi [Apiieup (?) 
Fried lei n ] Ka8o\uttitTtpa (watipfy. 

1 PrOClliS, p. 67, 11 TWf tiraixtluv roWa. ivtvp*. 

1 Matktmatutha i» Arisleitlts in Ad/wndtungen fur Gach. d. math. Wisstnschafttn, 
XVIII. Heft (1904), pp. 1—49. 

* Anal. pest. I. 6, 74 b 5. ' **»<£ I. 'O, 76 a 31 — 77 a 4. 



n8 INTRODUCTION [ch. ix. j 3 

principles peculiar to a science are the assumptions that a line is of 
such and such a character, and similarly for the straight (line); whereas 
it is a common principle, for instance, that, if equals be subtracted 
from equals, the remainders are equal. But it is enough that each of 
the common principles is true so far as regards the particular genus 
(subject-matter) ; for (in geometry) the effect will be the same even if 
the common principle be assumed to be true, not of everything, but 
only of magnitudes, and, in arithmetic, of numbers. 

" Now the things peculiar to the science, the existence of which 
must be assumed, are the things with reference to which the science 
investigates the essential attributes, e.g. arithmetic with reference to 
units, and geometry with reference to points and l : nes. With these 
things it is assumed that they exist and that they are of such and 
such a nature. But, with regard to their essential properties, what is 
assumed is only the meaning of each term employed : thus arithmetic 
assumes the answer to the question what is (meant by) ' odd ' or 
'even,' 'a square' or 'a cube,' and geometry to the question 
what is (meant by) ' the irrational ' or ' deflection ' or (the so-called) 
' verging ' (to a point) ; but that there are such things is proved by 
means of the common principles and of what has already been 
demonstrated. Similarly with astronomy. For every demonstrative 
science has to do with three things, (i) the things which are assumed 
to exist, namely the genus (subject-matter) in each case, the essential 
properties of which the science investigates, (2) the common axioms 
so-called, which are the primary source of demonstration, and (3) the 
properties with regard to which all that is assumed is the meaning of 
the respective terms used. There is, however, no reason why some 
sciences should not omit to speak of one or other of these things. 
Thus there need not be any supposition as to the existence of the 
genus, if it is manifest that it exists (for it is not equally clear that 
number exists and that cold and hot exist) ; and, with regard to the 
properties, there need be no assumption as to the meaning of terms if 
it is clear r just as in the common (axioms) there is no assumption as 
to what is the meaning of subtracting equals from equals, because it is 
well known. But none the less is it true that there are three things 
naturally distinct, the subject-matter of the proof, the things proved, 
and the (axioms) from which (the proof starts). 

"Now that which isperse necessarily true, and must necessarily be 
thought so, is not a hypothesis nor yet a postulate. For demon- 
stration has not to do with reasoning from outside but with the 
reason dwelling in the soul, j ust as is the case with the syllogism. 
It is always possible to raise objection to reasoning from outside, 
but to contradict the reason within us is not always possible. Now 
anything that the teacher assumes, though it is matter of proof, 
without proving it himself, is a hypothesis if the thing assumed is 
believed by the learner, and it is moreover a hypothesis, not abso- 
lutely, but relatively to the particular pupil ; but, if the same thing 
is assumed when the learner either has no opinion on the subject 
or is of a contrary opinion, it is a postulate. This is the difference 



ch. ix. §3] FIRST PRINCIPLES 119 

between 3 hypothesis and a postulate ; for a postulate is that which 
is rather contrary than otherwise to the opinion of the learner, or 
whatever is assumed and used without being proved, although matter 
for demonstration. Now definitions are not hypotheses, for they do 
not assert the existence or non-existence of anything, while hypotheses 
are among propositions. Definitions only require to be understood : 
a definition is therefore not a hypothesis, unless indeed it be asserted 
that any audible speech is a hypothesis. A hypothesis is that from 
the truth of which, if assumed, a conclusion can be established. Nor 
are the geometer's hypotheses false, as some have said : I mean those 
who say that ' you should not make use of what is false, and yet the 
geometer falsely calls the line which he has drawn a foot long when 
it is not, or straight when it is not straight' The geometer bases no 
conclusion on the particular line which he has drawn being that which 
he has described, but (he refers to) what is illustrated by the figures. 
Further, the postulate and every hypothesis are either universal or 
particular statements; definitions are neither" (because the subject 
is of equal extent with what is predicated of it). 

Every demonstrative science, says Aristotle, must start from in- 
demonstrable principles : otherwise, the steps of demonstration would 
be endless. Of these indemonstrable principles some are (a) common 
to all sciences, others are (0) particular, or peculiar to the particular 
science; (a) the common principles are the axioms, most commonly 
illustrated by the axiom that, if equals be subtracted from equals, the 
remainders are equal. Coming now to (6) the principles peculiar to 
the particular science which must be assumed, we have first the genus 
or subject-matter, the existence of which must be assumed, viz. magni- 
tude in the case of geometry, the unit in the case of arithmetic. Under 
this we must assume definitions of manifestations or attributes of the 
genus, e.g. straight lines, triangles, deflection etc. The definition in 
itself says nothing as to the existence of the thing defined : it only 
requires to be understood. But in geometry, in addition to the genus 
and the definitions, we have to assume the existence of a few primary 
things which are defined, viz. points and lines only : the existence 
of everything else, e.g. the various figures made up of these, as 
triangles, squares, tangents, and their properties, e.g, incommensur- 
ability etc., has to be proved (as it is proved by construction arid 
demonstration). In arithmetic we assume the existence of the unit: 
but, as regards the rest, only the definitions, e.g, those of odd, even, 
square, cube, are assumed, and existence has to be proved. We have then 
clearly distinguished, among the indemonstrable principles, axioms 
and definitions, A postulate is also distinguished from a hypothesis, 
the latter being made with the assent of the learner, the former 
without such assent or even in opposition to his opinion (though, 
strangely enough, immediately after saying this, Aristotle gives a 
wider meaning to "postulate" which would cover "hypothesis" as well, 
namely whatever is assumed, though it is matter for proof, and used 
without being proved). Heiberg remarks that there is no trace in 
Aristotle of Euclid's Postulates, and that " postulate" in Aristotle has 



iso INTRODUCTION [ch. ix. §3 

a different meaning. He seems to base this on the alternative 
description of postulate, indistinguishable from a hypothesis ; but, 
if we take the other description in which it is distinguished from a 
hypothesis as being an assumption of something which is a proper 
subject of demonstration without the assent or against the opinion of 
the learner, it seems to fit Euclid's Postulates fairly well, not only the 
fi rst th ree ( post u I at i ng t h re e con struc t i on s), bu t e m i n en tl y al so the other 
two, that all right angles are equal, and that two straight lines meeting 
a third and making the internal angles on the same side of it less than 
two right angles will meet on that side. Aristotle's description also 
seems to me to suit the "postulates" with which Archimedes begins 
his book On the equilibrium of planes, namely that equal weights balance 
at equal distances, and that equal weights at unequal distances do not 
balance but that the weight at the longer distance will prevail. 

Aristotle's distinction also between hypothesis and definition, and 
between hypothesis and axiom, is clear from the following passage : 
"Among immediate syllogistic principles, I call that a thesis which- 
it is neither possible to prove nor essential for any one to hold who 
is to learn anything ; but that which it is necessary for any one to 
hold who is to learn anything whatever is an axiom : for there are 
some principles of this kind, and that is the most usual name by 
which we speak of them. But, of theses, one kind is that which 
assumes one or other side of a predication, as, for instance, that 
something exists or does not exist, and this is a hypothesis ; the other, 
which makes no Such assumption, is a definition. For a definition is 
a thesis: thus the arithmetician posits (rlOcTat) that a unit is that 
which is indivisible in respect of quantity ; but this is not a hypo- 
thesis, since what is meant by a unit and the fact that a unit exists 
are different things 1 ." 

Aristotle uses as an alternative term for axioms "common (things)," 
to, Kowd, or "common opinions" (koiwiI Sofoi)> as in the following 
passages. " That, when equals are taken from equals, the remainders 
are equal is (a) common (principle) in the case of alt quantities, but 
mathematics takes a separate department (atroXa^ovo'a) and directs its 
investigation to some portion of its proper subject-matter, as e.g. lines 
or angles, numbers, or any of the other quantities*.' 1 "The common 
(principles), e.g. that one of two contradictories must be true, that 

equals taken from equals etc., and the like" " " With regard to the 

principles of demonstration, it is questionable whether they belong to 
one science or to several. By principles of demonstration I mean the 
common opinions from which all demonstration proceeds, e.g. that one 
of two contradictories must be true, and that it is impossible for the 
same thing to be and not be 4 ." Similarly "every demonstrative 
(science) investigates, with regard to some subject-matter, the essential 
attributes, starting from the common opinions'." We have then here, 
as Heiberg says, a sufficient explanation of Euclid's term for axioms, 

1 Anal. pest. I. a, 71a 14—14. ' Mttaph. 1061 b 19—14. 

* Anal. post. \. n, 77 a 30. * Mttaph. 996 b 36 — 30. 

5 Mttaph. 907 a so — 11. 



ch. ix. §3] FIRST PRINCIPLES lai 

viz. common notions (koivoI evvoiat), and there is no reason to suppose 
it to be a substitution for the original term due to the Stoics : cf. 
Proclus' remark that, according to Aristotle and the geometers, axiom 
and common notion are the same thing 1 . 

Aristotle discusses the indemonstrable character of the axioms 
in the Metaphysics. Since "all the demonstrative sciences use the 
axioms'," the question arises, to what science does their discussion 
belong*? The answer is that, like that of Being (oiJct-mi), it is the 
province of the (first) philosopher 1 . It is impossible that there should 
be demonstration of everything, as there would be an infinite series of 
demonstrations: if the axioms were the subject of a demonstrative 
science, there would have to be here too, as in other demonstrative 
sciences, a subject-genus, its attributes and corresponding axioms 1 \ thus 
there would be axioms behind axioms, and so on continually. The 
axiom is the most firmly established of all principles*. It is ignorance 
alone that could lead anyonetotryto prove the axioms' ; the supposed 
proof would be a.petilio principii'. If it is admitted that not every- 
thing can be proved, no one can point to any principle more truly 
indemonstrable'. If any one thought he could prove them, he could 
at once be refuted ; if he did not attempt to say anything, it would 
be ridiculous to argue with him i he would be no better than a 
vegetable 10 . The first condition of the possibility of any argument 
whatever is that words should signify something both to the speaker 
and to the hearer: without this there can be no reasoning with any one. 
And, if any one admits that words can mean anything to both hearer 
and speaker, he admits that something can be true without demon- 
stration. And so on". 

It was necessary to give some sketch of Aristotle's view of the 
first principles, if only in connexion with Proclus' account, which is 
as follows. As in the case of other sciences, so " the compiler of 
elements in geometry must give separately the principles of the 
science, and after that the conclusions from those principles, not 
giving any account of the principles but only of their consequences. 
No science proves its own principles, or even discourses about them : 
they are treated as self-evident. . .Thus the first essential was to dis- 
tinguish the principles from their consequences. Euclid carries out 
this plan practically in every book and, as a preliminary to the whole 
enquiry, sets out the common principles of this science. Then he 
divides the common principles themselves into hypotheses, postulates, 
and axioms. For all these are different from one another : an axiom, 
a postulate and a hypothesis are not the same thing, as the inspired 
Aristotle somewhere says. But, whenever that which is assumed and 
ranked as a principle is both known to the learner and convincing in 
itself, such a thing is an axiom, e.g. the statement that things which 
are equal to the same thing are also equal to one another. When, on 

1 Proclus, p. 194, 8. ' Mtlaph. 997 a 10. 

' ibid. 996 b 16, * ibid. 100s a 11 — b 11. ' Hid. 997 a 5 — 8. 

■ ibid. 1005 b II — 17. ' ibid, loofia 5. s ibid. 1006 a 17. 

ibid. 1006a 10. I0 ibid. 1006a 11 — 15. 1} ibid. 1006a iSsqq. 



u» INTRODUCTION [cH.1x.j3 

the other hand, the pupil has not the notion of what is told him 
which carries conviction in itself, but nevertheless lays it down and 
assents to its being assumed, such an assumption is a hypothesis. 
Thus we do not preconceive by virtue of a common notion, and 
without being taught, that the circle is such and such a figure, but, 
when we are told so, we assent without demonstration. When again 
what is asserted is both unknown and assumed even without the 
assent of the learner, then, he says, we call this a postulate, e.g. that 
all right angles are equal. This view of a postulate is clearly implied 
by those who have made a special and systematic attempt to show, 
with regard to one of the postulates, that it cannot be assented to by 
any one straight off. According then to the teaching of Aristotle, an 
axiom, a postulate and a hypothesis are thus distinguished 1 ." 

We observe, first, that Proclus in this passage confuses hypotheses 
and definitions, although Aristotle had made the distinction quite 
plain. The confusion may be due to his having in his mind a passage 
of Plato from which he evidently got the phrase about " not giving 
an account of" the principles. The passage is a : " I think you know 
that those who treat of geometries and calculations (arithmetic) and 
such things take for granted (inroQefievoi) odd and even, figures, 
angles of three kinds, and other things akin to these in each subject, 
implying that they know these things, and, though using them as 
hypotheses, do not even condescend to give any account of them 
either to themselves or to others, but begin from these things and 
then go through everything else in order, arriving ultimately, by 
recognised methods, at the conclusion which they started in search 
of." But the hypothesis is here the assumption, e.g, ' that there may 
he sttch a thing as length without breadth, henceforward called a line',' 
and so on, without any attempt to show that there is such a thing ; 
it is mentioned in connexion with the distinction between Plato's 
'superior' and 'inferior' intellectual method, the former of which 
uses successive hypotheses as stepping-stones by which it mounts 
upwards to the idea of Good. 

We pass now to Proclus' account of the difference between postu- 
lates and axioms. He begins with the view of Geminus, according 
to which " they differ from one another in the same way as theorems 
are also distinguished from problems. For, as in theorems we propose 
to see and determine what follows on the premisses, while in problems 
we are told to find and do something, in like manner in the axioms 
such things are assumed as are manifest of themselves and easily 
apprehended by our untaught notions, while in the postulates we 
assume such things as are easy to find and effect (our understanding 
suffering no strain in their assumption), and we require no complication 
of machinery*."..." Both must have the characteristic of being simple 

1 Proclos, pp. 7S, 10—77, *• 

' Republic, VI, 510 c. Cf. Aristotle, jVk. Eth. 1151a 17. 

* H. Jackson, Journal of Philology, vol. x. p. 144- 

* Proclus, pp. 178, 1? — (79,8* In illustration Proclus contrasts the drawing of a straight 
Line or a circle with the drawing of a " single- turn spiral " or of an equilateral triangle, the 



ch. ix. §3] FIRST PRINCIPLES 123 

and readily grasped, I mean both the postulate and the axiom ; but 
the postulate bids us contrive and find some subject-matter (uXj,) to 
exhibit a property simple and easily grasped, while the axiom bids us 
assert some essential attribute which is self-evident to the learner, 
just as is the fact that fire is hot, or any of the most obvious things 1 ," 

Again, says Proclus, " some claim that alt these things are alike 
postulates, in the same way as some maintain that all things that are 
sought are problems. For Archimedes begins his first book on /«- 
equilibrium' 1 with the remark ' I postulate that equal weights at equal 
distances are in equilibrium,' though one would rather call this an 
axiom. Others call them all axioms in the same way as some regard 
as theorems everything that requires demonstration'." 

" Others again will say that postulates are peculiar to geometrical 
subject-matter, while axioms are common to all investigation which 
is concerned with quantity and magnitude. Thus it is the geometer 
who knows that all right angles are equal and how to produce in 
a straight line any limited straight line, whereas it is a Common notion 
that things which are equal to the same thing are also equal to one 
another, and it is employed by the arithmetician and any scientific 
person who adapts the general statement to his own subject 1 ." 

The third view of the distinction between a postulate and an axiom 
is that of Aristotle above described'. 

The difficulties in the way of reconciling Euclid's classification 
of postulates and axioms with any one of the three alternative views 
are next dwelt upon. If we accept the first view according to which 
an axiom has reference to something known, and a postulate to 
something done, then the 4th postulate (that all right angles are 
equal) \% not a postulate ; neither is the 5th which states that, if a 
straight line falling on two straight lines makes the interior angles 
on the same side less than two right angles, the straight lines, if 
produced indefinitely, will meet on that side on which are the angles 
less than two right angles. On the second view, the assumption that 
two straight lines cannot enclose a space, " which even now," says 
Proclus, " some add as an axiom," and which is peculiar to the 
subject-matter of geometry, like the fact that all right angles are 
equal, is not an axiom. According to the third (Aristotelian) view, 
"everything which is confirmed (irt<not>Tat) by a sort of demonstration 

spiral requiring more complex machinery and even the equilateral triangle needing a certain 
method. " For the geometrical intelligence will say that hy conceiving a straight tine fined 
at one end but, as regards the other end, moving round the fixed end, and a point moving 
along the straight line from the fined end, 1 have described the single- turn spiral ; for the 
end of the straight line descrihing a circle, and the point moving on the straight line simul- 
taneously, when they arrive and meet at the same point, complete such a spiral. And again, 
if I draw equal circles, join their common point to the centres of the circles and draw a 
straight line from one of the centres to the other, I shall have the equilateral triangle. 
These things then are far from being completed by means of a single act or of a moment's 
thought" (p. 180. S— ji). 

1 Proclus, p. i Si, 4 — 11. 

1 It is necessary to coin a word to render tontapfxirtur, which is moreover in the plural. 
The title of the treatise as we have it is Equilibria of plana or centres of gravity of planes in 
Book r and Equilibria of planes in Book It. 

• Proclus, p. 181, 16—13. * **S p. 181, 6—14. ■ Pp. 118, 119. 



i»4 INTRODUCTION [ch. ix. §3 

will be a postulate, and what is incapable of proof will be an axiom 1 ." 
This last statement of Proclus is loose, as regards the axiom, because 
it omits Aristotle's requirement that the axiom should be a self- 
evident truth, and one that must be admitted by any one who is to 
learn anything at all, and, as regards the postulate, because Aristotle 
calls a postulate something assumed without proof though it is 
"matter of demonstration" (aTroSetierhv Sv), but says nothing of a 
quasi -demon strati on of the postulates. On the whole I think it is 
from Aristotle that we get the best idea of what Euclid understood 
by a postulate and an axiom or common notion. Thus Aristotle's 
account of an axiom as a principle common to all sciences, which is 
self-evident, though incapable of proof, agrees sufficiently with the 
contents of Euclid's common notions as reduced to five in the most 
recent text (not omitting the fourth, that " things which coincide are 
equal to one another"). As regards the postulates, it must be borne 
in mind that Aristotle says elsewhere' that, "other things being equal, 
that proof is the better which proceeds from the fewer postulates or 
hypotheses or propositions." I f then we say that a geometer must 
lay down as principles, first certain axioms or common notions, and 
then an irreducible minimum of postulates in the Aristotelian sense 
concerned only with the subject-matter of geometry, we are not far 
from describing what Euclid in fact does. As regards the postulates 
we may imagine him saying : ■ Besides the common notions there are 
a few other things which I must assume without proof, but which 
differ from the common notions in that they are not self-evident. 
The learner may or may not be disposed to agree to them ; but he 
must accept them at the outset on the superior authority of his 
teacher, and must be left to convince himself of their truth in the 
course of the investigation which follows. In the first place certain 
simple constructions, the drawing and producing of a straight line, 
and the drawing of a circle, must be assumed to be possible, and with 
the constructions the existence of such things as straight lines and 
circles ; and besides this we must lay down some postulate to form 
the basis of the theory of parallels." It is true that the admission of 
the 4th postulate that all right angles are equal still presents a 
difficulty to which we shall have to recur. 

There is of course no foundation for the idea, which has found 
its way into many text-books, that " the object of the postulates is to 
declare that the only instruments the use of which is permitted in 
geometry are the rule and compass'." 



§ 4. THEOREMS AND PROBLEMS. 

" Again the deductions from the first principles," says Proclus, 
"are divided into problems and theorems, the former embracing the 

1 Proclus, pp. i8j, 11—183, '3- * Anal. past. 1. aj, 86* 33—35. 

8 Cf. Lardner's Euclid : aha Todhunter. 



ch. ix. 5 4 ] THEOREMS AND PROBLEMS i*5 

generation, division, subtraction or addition of figures, and generally 
the changes which are brought about in them, the latter exhibiting 
the essential attributes of each 1 ," 

" Now, of the ancients, some, like Speusippus and Amphinomus, 
thought proper to call them all theorems, regarding the name of 
theorems as more appropriate than that of problems to theoretic 
sciences, especially as these deal with eternal objects. For there is 
no becoming in things eternal, so that neither could the problem 
have any place with them, since it promises the generation and 
making of what has not before existed, e.g. the construction of an 
equilateral triangle, or the describing of a square on a given straight 
line, or the placing of a straight line at a given point. Hence they 
say it is better to assert that all (propositions) are of the same kind, 
and that we regard the generation that takes place in them as 
referring not to actual making but to knowledge, when we treat things 
existing eternally as if they were subject to becoming: in other words, 
we may say that everything is treated by way^ of theorem and not 
by way of problem* (irdvra 6eapr)fjiaTiKw oX\* ov TrpQJ3\iifiaTtic<io<i 
\etfi0 ui' hit (tat). 

" Others on the contrary, like the mathematicians of the school 
of Menaechmus, thought it right to call them all problems, describing 
their purpose as twofold, namely in some cases to furnish (yropi- 
ffatr&at) the thing sought, in others to take a determinate object 
and see either what it is, or of what nature, or what is its property, 
or in what relations it stands to something else. 

"In reality both assertions are correct. Speusippus is right 
because the problems of geometry are not like those of mechanics, 
the latter being matters of sense and exhibiting becoming and change 
of every sort. The school of Menaechmus are right also because the 
discoveries even of theorems do not arise without an issuing- forth 
into matter, by which I mean intelligible matter. Thus forms going 
out into matter and giving it shape may fairly be said to be like 
processes of becoming. For we say that the motion of our thought 
and the thro wing-out of the forms in it is what produces the figures 
in the imagination and the conditions subsisting in them. It is in 
the imagination that constructions, divisions, placings, applications, 
additions and subtractions (take place), but everything in the mind is 
fixed and immune from becoming and from every sort of change 1 ." 

" Now those who distinguish the theorem from the problem say 
that every problem implies the possibility, not only of that which is 
predicated of its subject-matter, but also of its opposite, whereas 
every theorem implies the possibility of the thing predicated but not 
of its opposite as well. By the subject-matter I mean the genus 
which is the subject of inquiry, for example, a triangle or a square 
or a circle, and by the property predicated the essential attribute, 
as equality, section, position, and the like. When then any or-e 

' Proclus, p, 77, 7— n. * Hid. pp. 77, 15—78, 8. 

* iHd, pp. j8, S — 79, 1. 



i26 INTRODUCTION [ch. ix, $4 

enunciates thus, To inscribe an equilateral triangle in a circle, he states 
a problem ; for it is also possible to Inscribe in it a triangle which 
is not equilateral. Again, if we take the enunciation On a given 
limited straight line to construct an equilateral triangle, this is a 
problem ; for it is possible also to construct one which is not equi- 
lateral. But, when any one enunciates that In isosceles triangles tlu 
angles at the base are equal, we must say that he enunciates a theorem ; 
for it is not also possible that the angles at the base of isosceles 
triangles should be unequal. It follows that, if any one were to use 
the form of a problem and say In a semicircle to describe a right angle, 
he would be set down as no geometer. For every angle in a semi- 
circle is right 1 .'' 

'' Zenodotus, who belonged to the succession of Oenoptdes, but 
was a disciple of Andron, distinguished the theorem from the problem 
by the fact that the theorem inquires what is the property predicated 
of the subject-matter in it, but the problem what is the cause of what 
effect (rivoi Sirros ri icrriv). Hence too Posidonius denned the one 
(the problem) as a proposition in which it is inquired whether a thing 
exists or not (et ffo-rte t) /*■$), the other (the theorem*) as a proposition 
in which it is inquired what (a thing) is or of what nature (ri itrrtv fj 
•irolov Tt) ; and he said that the theoretic proposition must be put in a 
declaratory form, e.g., Any triangle has two sides (together) greater than 
the remaining side and In any isosceles triangle the angles at the base 
are equal, but that we should state the problematic proposition as If 
inquiring whether it is possible to construct an equilateral triangle 
upon such and such a straight line. For there is a difference between 
inquiring absolutely and indeterminately (a-n-Xaq re KaX dopl<rrto<i) 
whether there exists a straight line from such and such a point at 
right angles to such and such a straight line and investigating which 
is the straight line at right angles*." 

"That there is a certain difference between the problem and the 
theorem is clear from what has been said ; and that the Elements of 
Euclid contain partly problems and partly theorems will be made 
manifest by the individual propositions, where Euclid himself adds at 
the end of what is proved in them, in some cases, 'that which it was 
required to do,' and in others, ' that which it was required to prove, 1 
the latter expression being regarded as characteristic of theorems, in 
spite of the fact that, as we have said, demonstration is found in 
problems also. In problems, however, even the demonstration is for 
the purpose of (confirming) the construction : for we bring in the 
demonstration in order to show that what was enjoined has been 
done ; whereas in theorems the demonstration is worthy of study for 
its own sake as being capable of putting before us the nature of the 
thing sought. And you will find that Euclid sometimes interweaves 
theorems with problems and employs them in turn, as in the first 

* Pioclus, pp. 79, n^So, 5. 

* In the text we have ri &t rpiflX^fia answering to ri pin without substantive : Trp&p\i)jin 
was obvious]; inserted in error, 

* Prochis, pp. 80, is — 8i, 4. 



ch. ix. 54] THEOREMS AND PROBLEMS ia? 

book, while at other times he makes one or other preponderate. 
For the fourth book consists wholly of problems, and the fifth of 
theorems 1 ." 

Again, in his note on Eucl. I. 4, Proclus says that Carpus, the 
writer on mechanics, raised the question of theorems and problems in 
his treatise on astronomy. Carpus, we are told, " says that the class 
of problems is in order prior to theorems. For the subjects, the 
properties of which are sought, are discovered by means of problems. 
Moreover in a problem the enunciation is simple and requires no 
skilled intelligence; it orders you plainly to do such and such a 
thing, to construct an equilateral triangle, or, given two straight lines, to 
cut off from the greater (a straight line) equal to the lesser, and what is 
there obscure or elaborate in these things ? But the enunciation of a 
theorem is a matter of labour and requires much exactness and 
scientific judgment in order that it may not turn out to exceed or 
fail short of the truth ; an example is found even in this proposition 
(I. 4), the first of the theorems. Again, in the case of problems, one 
general way has been discovered, that of analysis, by following which 
we can always hope to succeed ; it is this method by which the more 
obscure problems are investigated. But, in the case of theorems, the 
method of setting about them is hard to get hold of since ' up to our 
time,' says Carpus, ' no one has been able to hand down a general 
method for their discovery. Hence, by reason of their easiness, the 
class of problems would naturally be more simple/ After these 
distinctions, he proceeds : ' Hence it is that in the Elements too 
problems precede theorems, and the Elements begin from them ; the 
first theorem is fourth in order, not because the fifth 11 is proved from 
the problems, but because, even if it needs tor its demonstration none 
of the propositions which precede it, it was necessary that they should 
be first because they are problems, while it is a theorem. In fact, in 
this theorem he uses the common notions exclusively, and in some 
sort takes the same triangle placed in different positions ; the 
coincidence and the equality proved thereby depend entirely upon 
sensible and distinct apprehension. Nevertheless, though the demon- 
stration of the first theorem is of this character, the problems properly 
preceded it, because in general problems are allotted the order of 
precedence*.'" 

Proclus himself explains the position of Prop. 4 after Props, i- — 3 
as due to the fact that a theorem about the essential properties of 
triangles ought not to be introduced before we know that such a 
thing as a triangle can be constructed, nor a theorem about the 
equality of sides or straight lines until we have shown, by constructing 
them, that there can be two straight lines which are equal to one 
another 4 . It is plausible enough to argue in this way that Props. 2 
and 3 at all events should precede Prop, 4. And Prop, 1 is used in 

1 Proclus, p. Si, 5 — is. 

9 ri> ri^Trzav. This should apparently be the fourth because in the next words it is 
implied that none of the first three propositions ate required in proving it. 
* Proclus, pp. 141, it) — 543, 11. * ibid. pp. 133, 11— 1134, 6. 



1*8 INTRODUCTION [ch. ix. §4 

Prop. 2, and must therefore precede it But Prop. 1 showing how to 
construct an equilateral triangle on a given base is not important, in 
relation to Prop. 4, as dealing with the " production of triangles " in 
general : for it is of no use to say, as Proclus does, that the construc- 
tion of the equilateral triangle is " common to the three species (of 
triangles) 1 ," as we are not in a position to know this at such an early 
stage. The existence of triangles in general was doubtless assumed as 
following from the existence of straight lines and points in one plane 
and from the possibility of drawing a straight line from one point to 
another. 

Proclus does not however seem to reject definitely the view of 
Carpus, for he goes on* : "And perhaps problems are in order before 
theorems, and especially for those who need to ascend from the arts 
which are concerned with things of sense to theoretical investigation. 
But in dignity theorems are prior to problems.... It is then foolish to 
blame Geminus for saying that the theorem is more perfect than the 
problem. For Carpus himself gave the priority to problems in respect 
of order, and Geminus to theorems in point of more perfect dignity," 
so that there was no real inconsistency between the two. 

Problems were classified according to the number of their possible 
solutions. Amphinomus said that those which had a unique solution 
(itowa^tSt) were called " ordered " (the word has dropped out in 
Proclus, but it must be rerayftiva, in contrast to the third kind, 
aTattra) ; those which had a definite number of solutions " inter- 
mediate" (fiea-a); and those with an infinite variety of solutions "un- 
ordered " (ajaMTa)'. Proclus gives as an example of the last the 
problem To divide a given straight line into three parts in continued 
proportion*. This is the same thing as solving the equations %+y+s=a, 
xs =_y i . Proclus' remarks upon the problem show that it was solved, 
like all quadratic equations, by the method of " application of areas." 
The straight line a was first divided into any two parts, (jr-M)and_y, 
subject to the sole limitation that (x + s) must not be less than 2y, 
which limitation is the Stopto-fws, or condition of possibility. Then 
an area was applied to (x + z), or («— y), "falling short by a square 
figure" {eWetirov tXhei rerpaywy) and equal to the square on y. This 
determines x and z separately in terms of a and y. For, if b be the 
side of the square by which the area (i.e. rectangle) "falls short," we 
have {{a —y) — z\z ™j**, whence 2z «= (a —y) ± n/[(a —yf — 4y 3 }. And 
y may be chosen arbitrarily, provided that it is not greater than 0/3. 
Hence there are an infinite number of solutions. If y = a(i, then, as 
Proclus remarks, the three parts are equal. 

Other distinctions between different kinds of problems are added 
by Proclus. The word " problem," he says, is used in several senses. 
In its widest sense it may mean anything " propounded " (irpojewo- 
pi-i>ov), whether for the purpose of instruction {paBrio-em) or construc- 
tion (■n-oojo'fci)?). (In this sense, therefore, it would include a theorem.) 

1 Proclus, p. 13^, Ji. ■ ibid, p. 543, 11—15. 

■ ibid, p. »jo> 7 — 13. * ibid. pp. no, 16 — 111, 6. 



ch. ix. §4] THEOREMS AND PROBLEMS 139 

But its special sense in mathematics is that of something "propounded 
with a view to a theoretic construction 1 ." 

Again you may apply the term (in this restricted sense) even to 
something which is impossible, although it is more appropriately used 
of what is possible and neither asks too much nor contains too little in 
the shape of data. According as a problem has one or other of these 
defects respectively, it is called (1) a problem in excess (irXeovdgov) or 
(2) a deficient problem (iK\nre<; irp60\T}p.a), The problem in excess 
(1) is of two kinds, (a) a problem in which the properties of the 
figure to be found are either inconsistent (atrii p/Sara) or non-existent 
{avvirapicTa), in which case the problem is called impossible, or (b) a 
problem in which the enunciation is merely redundant : an example 
of this would be a problem requiring us to construct an equilateral 
triangle with its vertical angle equal to two-thirds of a right angle ; 
such a problem is possible and is called "more than a problem '' (p.el&v 
7j Trp6ft\T)fta). The deficient problem (2) is similarly called " less than 
a problem " (tKaatrov rj Trpdft\i)f*a), its characteristic being that 
something has to be added to the enunciation in order to convert it 
from indeterminateness (aopttr-ria) to order (raf «) and scientific deter- 
minateness (Spas evta-TtjtioviKos) : such would be a problem bidding 
you " to construct an isosceles triangle," for the varieties of isosceles 
triangles are unlimited. Such "problems" are not problems in the 
proper sense {/cupim<! \cy6ftei/a vpofiXjjpaTa), but only equivocally". 

§ 5. THE FORMAL DIVISIONS OF A PROPOSITION. 

"Every problem," says Proclus', "and every theorem which is 
complete with all its parts perfect purports to contain in itself all of 
the following elements: enunciation (TrpoVao-is), setting-out (e*0Wt<:), 
definition or specification (Stop tvr fiat), construction or machinery 
(xaratrKt utj), proof (airoSeifiv), conclusion (uvp/!repai7jt,a). Now of 
these the enunciation states what is given and what is that which is 
sought, the perfect enunciation consisting of both these parts. The 
setting-out marks off what is given, by itself, and adapts it before- 
hand for use in the investigation. The definition or specification 
states separately and makes clear what the particular thing is which 
is sought. The construction or machinery adds what is wanting to the 
datum for the purpose of finding what is sought .The proof draws 
the required inference by reasoning scientifically from acknowledged 
facts. The conclusion reverts again to the enunciation, confirming 
what has been demonstrated. These are all the parts of problems 
and theorems, but the most essential and those which are found in all 
are enunciation, proof, conclusion. For it is equally necessary to know 
beforehand what is sought, to prove this by means of the intermediate 
steps, and to state the proved fact as a conclusion ; it is impossible 
to dispense with any of these three things. The remaining parts 
are often brought in, but are often left out as serving no purpose. 

1 proclus, p. aii, J— 1 r. a ibid. pp. Mr, 13 — 111, i+. 

* ibid. pp. 503, i — ]o+, 13 ; 1O4, 13 — 105, 8. 



130 INTRODUCTION [ch. rx. js 

Thus there is neither setting-out nor definition in the problem of 
constructing an isosceles triangle having each of the angles at the 
base double of the remaining angle, and in most theorems there 
is no construction because the setting'Oiit suffices without any addition 
for proving the required property from the data. When then do 
we say that the setting-out is wanting ? The answer is, when there 
is nothing given in the enunciation ; for, though the enunciation is 
in general divided into what is given and what is sought, this 
is not always the case, but sometimes it states only what is sought, 
i.e. what must be known or found, as in the case of the problem 
just mentioned. That problem does not, in fact, state beforehand 
with what datum we are to construct the isosceles triangle having 
each of the equal angles double of the remaining angle, but (simply) 
that we are to find such a triangle When, then, the enuncia- 
tion contains both (what is given and what is sought), in that case 
we find both definition and setting-out, but, whenever the datum 
is wanting, they too are wanting. For not only is the setting-out 
concerned with the datum, but so is the definition also, as, in the 
absence of the datum, the definition will be identical with the 
enunciation. In fact, what could you say in defining the object of 
the aforesaid problem except that it is required to find an isosceles 
triangle of the kind referred to ? But that is what the enunciation 
stated. If then the enunciation does not include, on the one hand, 
what is given and, on the other, what is sought, there is no setting-out 
in virtue of there being no datum, and the definition is left out in 
order to avoid a mere repetition of the enunciation." 

The constituent parts of an Euclidean proposition will be readily 
identified by means of the above description. As regards the defi- 
nition or specification (S*opt<r/ios) it is to be observed that we have 
here only one of its uses. Here it means a closer definition or descrip- 
tion of the object aimed at, by means of the concrete lines or figures 
set out in the eK&eai<; instead of the general terms used in the enun- 
ciation ; and its purpose is to rivet the attention better, as Proclus 
indicates in a later passage (rpoirov rtvk irpotrextia? itrrlv aiVioe o 

The other technical use of the word to signify the limitations to 
which the possible solutions of a problem are subject is also described 
by Proclus, who speaks of hiopio-fioi determining " whether what is 
sought is impossible or possible, and how far it is practicable and in 
how many ways 1 " ; and the Biopur/ioit in this sense appears in Euclid 
as well as in Archimedes and Apollonius. Thus we have in Eucl. I, 
22 the enunciation " From three straight lines which are equal to 
three given straight lines to construct a triangle," followed imme- 
diately by the limiting condition ($tap«rfi6<;). " Thus two of the 
straight lines taken together in any manner must be greater than the 
remaining one." Similarly in VI. 28 the enunciation "To a given 
straight line to apply a parallelogram equal to a given rectilineal 

1 Proclus, p. ?o8, ?i, * ibid. \i. loa, 3. 



ch. ix. f 5] FORMAL DIVISIONS OF A PROPOSITION 131 

figure and falling short by a parallelogrammic figure similar to a 
given one " is at once followed by the necessary condition of possi- 
bility: "Thus the given rectilineal figure must not be greater than 
that described on half the line and similar to the defect." 

Tannery supposed that, in giving the other description of the 
Biopta/tos as quoted above, Proclus, or rather his guide, was using the 
term incorrectly. The Btopta-fios in the better known sense of the 
determination of limits or conditions of possibility was, we are told, 
invented by Leon. Pappus uses the word in this sense only. The 
other use of the term might, Tannery thought, be due to a confusion 
occasioned by the use of the same words (Bu &}) in introducing the 
parts of a proposition corresponding to the two meanings of the word 
Btopur/io^ 1 . On the other hand it is to be observed that Eutocius 
distinguishes clearly between the two uses and implies that the differ- 
ence was well known 1 . The SiopuTft.6^ in the sense of condition of 
possibility follows immediately on the enunciation, is even part of it ; 
the &iopitry.6s in the other sense of course comes immediately after the 
setting-out. 

Proclus has a useful observation respecting the conclusion of a 
proposition 3 . " The conclusion they are accustomed to make double 
in a certain way : 1 mean, by proving it in the given case and then 
drawing a general inference, passing, that is, from the partial con- 
clusion to the general. For, inasmuch as they do not make use of 
the individuality of the subjects taken, but only draw an angle or a 
straight line with a view to placing the datum before our eyes, they 
consider that this same fact which is established in the case of the 
particular figure constitutes a conclusion true of every other figure of 
the same kind. They pass accordingly to the general in order that 
we may not conceive the conclusion to be partial. And they are 
justified in so passing, since they use for the demonstration the par- 
ticular things set out, not qud particulars, but qua typical of the rest 
For it is not in virtue of such and such a size attaching to the angle 
which is set out that I effect the bisection of it, but in virtue of its 
being rectilineal and nothing more. Such and such size is peculiar to 
the angle set out, but its quality of being rectilineal is common to all 
rectilineal angles. Suppose, for example, that the given angle is a 
right angle. If then I had employed in the proof the fact of its being 
right, I should not have been able to pass to every species of recti- 
lineal angle ; but, if I make no use of its being right, and only consider 
it as rectilineal, the argument will equally apply to rectilineal angles 
in general." 

L La Giotnltru graque, p. 149 note. Where 5ti 5% introduces the closer description of 
the pioblem we may translate, "it is then requited'' or "thus it is required" (to construct etc); 
when it introduces the condition of possibility we may translate "thus it is necessary etc." 
Heiberg originally wrote Sti 9i in the latter sense in I. 21 on the authority of Proclus and 
Eutocius, and against that of the mss. Later, on the occasion of XI. 13, he observed that he 
should have followed the MSS. and written Sti i5?j which he found to be, after all, the right 
reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). Je i Hi is also the expression used 
by Diophantus for introducing conditions of possibility. 

a See the passage of Eutocius referred to in last note. 3 Proclus, p. 307, 4 — 1$. 



i 3 a INTRODUCTION [ch. ix. §6 

I 6. OTHER TECHNICAL TERMS. 

I. Things said to be given. 

Proclus attaches to his description of the formal divisions of a 
proposition an explanation of the different senses in which the word 
give ft or datum (SeSo^evoi') is used in geometry. " Everything that is 
given is given in one or other of the following ways, in position, in 
ratio, in tnagnitude, or in species. The point is given in position only, 
but a line and the rest may be given in all the senses 1 ." 

The illustrations which Proclus gives of the four senses in which a 
thing may be given are not altogether happy, and, as regards things 
which are given in position, in magnitude, and in species, it is best, I 
think, to follow the definitions given by Euclid himself in his book of 
Data. Euclid does not mention the fourth class, things given in ratio, 
nor apparently do any of the great geometers. 

(i) Given in position really needs no definition ; and, when Euclid 
says {Data, Def 4) that " Points, lines and angles are said to be given 
in position which always occupy the same place," we are not really 
the wiser. 

(3) Given in tnagnitude is defined thus {Data, Def. 1): "Areas, 
lines and angles are called given in magnitude to which we can find 
equals." Proclus' illustration is in this case the following : when, he 
says, two unequal straight lines are given from the greater of which 
we have to cut off a straight line equal to the lesser, the straight lines 
are obviously given in magnitude, " for greater and less, and finite 
and infinite are predications peculiar to magnitude." But he does not 
explain that part of the implication of the term is that a thing is given 
in magnitude only, and that, for example, its position is not given and 
is a matter of indifference 

(3) Given in species. Euclid's definition {Data, Def. 3) is: 
" Rectilineal figures are said to be given in species in which the angles 
are severally given and the ratios of the sides to one another are 
given." And this is the recognised use of the term (cf. Pappus, 
passim) Proclus uses the term in a much wider sense for which I am 
not aware of any authority. Thus, he says, when we speak of (bisect- 
ing) a given rectilineal angle, the angle is given in species by the word 
rectilineal, which prevents our attempting, by the same method, to 
bisect a curvilineal angle I On Eucl. 1. 9, to which he here refers, he 
says that an angle is given in species when e.g. we say that it is right 
or acute or obtuse or rectilineal or " mixed," but that the actual angle 
in the proposition is given in species only. As a matter of fact, we 
should say that the actual angle in the figure of the proposition is 
given in magnitude and not in species, part of the implication of given 
in species being that the actual magnitude of the thing given in species 
is indifferent ; an angle cannot be given in species in this sense at all. 
The confusion in Proclus' mind is shown when, after saying that a 
right angle is given in species, he describes a third of a right angle as 
given in magnitude. 

1 Proclus, p, Wj, 13—15. 



ch. ix. §6] OTHER TECHNICAL TERMS 133 

No better example of what is meant by given in species, in its 
proper sense, as limited to rectilineal figures, can be quoted than the 
given parallelogram in Eucl. VI. 28, to which the required parallelo- 
gram has to be made similar ; the former parallelogram is in fact 
given in species, though its actual size, or scale, is indifferent 

(4) Given in ratio presumably means something which is given 
by means of its ratio to some other given thing. This we gather from 
Proclus' remark (in his note on 1. 9) that an angle may be given in 
ratio " as when we say that it is double and treble of such and such an 
angle or, generally, greater and less." The term, however, appears to 
have no authority and to serve no purpose. Proclus may have 
derived it from such expressions as "in a given ratio" which are 
common enough. 

2. Lemma. 

"The term lemma" says Proclus 1 , "is often used of any proposition 
which is assumed for the construction of something else : thus it is a 
common remark that a proof has been made out of such and such 
lemmas. But the special meaning of lemma in geometry is a 
proposition requiring confirmation. For when, in either construction 
or demonstration, we assume anything which has not been proved but 
requires argument, then, because we regard what has been assumed as 
doubtful in itself and therefore worthy of investigation, we call it a 
lemma*, differing as it does from the postulate and the axiom in being 
matter of demonstration, whereas they are immediately taken for 
granted, without demonstration, for the purpose of confirming other 
things. Now in the discovery of lemmas the best aid is a mental 
aptitude for it. For we may see many who are quick at solutions and 
yet do not work by method ; thus Cratistus in our time was able to 
obtain the required result from first principles, and those the fewest 
possible, but it was his natural gift which helped him to the discovery. 

1 Proclus, pp. 111, 1 — m, 4. 

* It would appear, says Tannery (p. 1 ji «.), that Gem in us understood a lemma as being 
simply \atxflwt6f££vov t something assumed (cf. Lhe passage of Proclus, p. 73, 4, relating to 
Mcnaechmus* vie* of eUmertis) : hence we cannot consider ourselves authorised in attributing 
to Geminus the more technical definition of the term here given by Proclus, according to 
which it is only used of propositions not proved beforehand. This view of a lemma must 
be considered as relatively modern. It seems to have had its origin in an imperfection of 
method. In the course of a demonstration it was necessary to assume a proposition which 
required proof, but the proof of which would, if inserted in lhe particular place, break the 
thread of the demonstration : hence it was necessary either to prove it beforehand as a 
preliminary proposition or to postpone it to be proved afterwards {<Sit i& inx^irtrai). 
when, after the time of Ge minus, the progress of original discovery in geometry was arrested, 
geometers occupied themselves with the study and elucidation of the works of the great 
mathematicians who had preceded them. This involved the investigation of propositions 
explicitly quoted or tacitly assumed in the great classical treatises; and naturally it was found 
that several such remained to be demonstrated, either because Lhe authors had omitted 
them as being easy enough to be left to the reader himself to prove, or because books in 
which they were proved had been lost in the meantime. Hence arose a class of complementary 
or auxiliary propositions which were called lemmas. Thus Pappus gives in his Book VII a 
collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the 
so-called "Treasury of Analysis " (r&rot AvtOwdtxevat), When Procltis goes on to distinguish 
three methods of discovering lemmas, analysis, division, and redurtio ad absurdum, he seems 
to imply that the principal business of contemporary geometers was the investigation of these 
auxiliary propositions. 



i 3 4 INTRODUCTION [ch. ix. §6 

Nevertheless certain methods have been handed down. The finest is 
the method which by means of analysis carries the thing sought up to 
an acknowledged principle, a method which Plato, as they say, com- 
municated to Leodamas 1 , and by which the latter, too, is said to have 
discovered many things in geometry. The second is the method of 
division*, which divides into its parts the genus proposed for con- 
sideration and gives a starting-point for the demonstration by means 
of the elimination of the other elements in the construction of what is 
proposed, which method also Plato extolled as being of assistance to 
all sciences. The third is that by means of the reductio ad absurdntn, 
which does not show what is sought di recti yi but refutes its opposite 
and discovers the truth incidentally.'' 

3. Case. 

"The case* (irTmtTK)," Proclus proceeds*, "announces different ways 
of construction and alteration of positions due to the transposition of 
points or lines or planes or solids. And, in general, all its varieties 
are seen in the figure, and this is why it is called case, being a trans- 
position in the construction." 

4. Porism. 

" The term porism is used also of certain problems such as the 
Porisms written by Euclid. But it is specially used when from what 
has been demonstrated some other theorem is revealed at the same 
time without our propounding it, which theorem has on this very 
account been called a porism (corollary) as being a sort of incidental 
gain arising from the scientific demonstration ." Cf. the note on 1. 15, 

1 This passage and another from Diogenes Laertiiis (ur. 54, p. 74 ed. Coliet) to the effect 
that "He [Plato] explained [tttnj-rfifaTO} to Leodamas of Thasos the method of inquiry by 
analysis " have been commonly understood as ascribing to Plato the invention of the method 
of analysis ; but Tannery points out forcibly (pp. 1 13, 113) bow difficult it is to explain in 
what Plato's discovery could have consisted if analysis be taken in the sense attributed to it 
in Pappus, where we can see no more than a series of successive, reductions of a problem 
until it is finally reduced to a known problem. On the other hand, Proclus' words about 
carrying up the thing sought to " an acknowledged principle " suggest that what he had in 
mind was the process described at the end of Book vr of the Republic by which the dialec- 
tician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which 
is not hypothetical, and then is able to descend step by step verifying every one of the 
hypotheses by which he ascended. This description does not of course refer to mathematical 
analysis, but it may have given rise to the idea that analysis was Plato's discovery, since 
analysis and synthesis following each other are related in the same way as the upward and 
the downward progression in the dialectician's intellectual method- And it may be that 
Plato's achievement was to observe the importance, from the point of view of logical rigour, 
of the confirmatory synthesis following analysis, and to regularise in this way and elevate 
into a completely irrefragable method the partial and uncertain analysis upon which the 
works of his predecessors depended. 

* Here again the successive biparlilions of genera into species such as we find in the 
Sephist and Republic have very little to say to geometry, and the very fact that they are here 
mentioned side by side with analysis suggests that Proclus confused the latter with the 
philosophical method of Hep. vi. 

3 Tannery rightly remarks (p. 151) that the subdivision of a theorem or problem into 
several cases is foreign to the really classic form ; the ancients preferred, where necessary, to 
multiply enunciations. As, however, some omissions necessarily occurred, the writers of 
lemmas naturally added separate cases, which in some instances found their way into the text. 
A good example is Euclid I. 7, the second case of which, as it appears in our text- books, 
was interpolated. On the commentary of Proclus on this proposition Th, Taylor rightly 
remarks that " Euclid everywhere avoids a multitude of cases." 

* Proclus, p. 11a, 5 — 11. 

a Tannery notes however that, so far from distinguishing his corollaries from the con- 



ch. ix. §6] OTHER TECHNICAL TERMS 135 

5. Objection. 

" The objection (eWrao-i?) obstructs the whole course of the argu- 
ment by appearing as an obstacle (or crying ' halt,' d-n-avrao-a) either 
to the construction or to the demonstration. There is this difference 
between the objection and the case, that, whereas he who propounds 
the case has to prove the proposition to be true of it, he who makes 
the objection does not need to prove anything : on the contrary it is 
necessary to destroy the objection and to show that its author is 
saying what is false'." 

That is, in general the objection endeavours to make it appear that 
the demonstration is not true in every case ; and it is then necessary 
to prove, in refutation of the objection, either that the supposed case 
is impossible, or that the demonstration is true even for that case. A 
good instance is afforded by EucL I. 7, The text- books give a second 
case which is not in the original text of Euclid. Proclus remarks on 
the proposition as given by Euclid that the objection may conceivably 
be raised that what Euclid declares to be impossible may after all be 
possible in the event of one pair of stiaight lines falling completely 
within the other pair. Proclus then refutes the objection by proving 
the impossibility in that case also. His proof then came to be given 
in the text-books as part of Euclid's proposition. 

The objection is one of the technical terms in Aristotle's logic and 
its nature is explained in the Prior Analytics''. "An objection is a 
proposition contrary to a proposition.,.. Objections are of two sorts, 
general or partial.... For when it is maintained that an attribute 
belongs to every (member of a class), we object either that it belongs 
to none (of the class) or that there is some one (member of the class) 
to which it does not belong." 

6. Reduction. 

This is again an Aristotelian term, explained in the Prior 
Analytics*. It is well described by Proclus in the following passage : 

" Reduction (diraywyq) is a transition from one problem or theorem 
to another, the solution or proof of which makes that which is pro- 
pounded manifest also. For example, after the doubling of the cube 
had been investigated, they transformed the investigation into another 
upon which it follows, namely the finding of the two means ; and from 
that time forward they inquired how between two given straight lines 
two mean proportionals could be discovered. And Tney say that the 
first to effect the reduction of difficult constructions was Hippocrates of 
Chios, who also squared a lune and discovered many other things in 
geometry, being second to none in ingenuity as regards constructions'." 

elusions of his propositions, Euclid inserts them before the closing words " (being) what it 
was required to do'' or M to prove." In fact the porism- corollary is with Euclid rather a 
modified form of the regular conclusion than a separate proposition. 
1 Proclus, p. an, 18 — 33. 

* Anal, prior. 11. 16, 69 a 37.- * ibid. 11. 15, 69 a 30. 

* Proclus, pp. an, 34— 113, 11. This passage has frequently been taken as crediting 
Hippocrates with the discovery of the method of geometrical reduction: ef. Taylor (Transla- 
tion of Proclus, 11. p. 16), Allman (p. 41 4., 50), Gow (pp. 169, 170), As Tannery remarks 
(p. no), if the particular reduction of the duplication problem to that of the two means is 



136 INTRODUCTION [ch. IX. {6 

7. Re duct io ad absurdum. 

This is variously called by Aristotle " reductio ad absurdum" ft) «*s 
rr) aZvvarav airaycoyi})', " proof per impossible " (17 Bia toS dZvvarov 
Setfw or a-n-o&etfcy, " proof leading to the impossible " (r) Wc to 
rtSufaroy ayova-a airoBeti is) 1 . It is part of " proof (starting) from a 
hypothesis' " («'f tiTrofeo-ew?). " All (syllogisms) which reach the 
conclusion per impossibile reason out a conclusion which is false, and 
they prove the original contention (by the method starting) from a 
hypothesis, when something impossible results from assuming the 
contradictory of the original contention, as, for example, when it is 
proved that the diagonal (of a square) is incommensurable because, 
if it be assumed commensurable, it will follow that odd (numbers) 
are equal to even (numbers) 1 ." Or again, "proof (leading) to the 
impossible differs from the direct (£ec*r(«ri}t) in that it assumes what 
it desires to destroy [namely the hypothesis of the falsity of the 
conclusion] and then reduces it to something admittedly false, whereas 
the direct proof starts from premisses admittedly true'." 

Proclus has the following description of the reductio ad absurdum. 
" Proofs by reductio ad absurdum in every case reach a conclusion 
manifestly impossible, a conclusion the contradictory of which is 
admitted. In some cases the conclusions are found to conflict with 
the common notions, or the postulates, or the hypotheses (from which 
we started); in others they contradict propositions previously estab- 
lished 7 ". .."Every reductio ad absurdum assumes what conflicts with 
the desired result, then, using that as a basis, proceeds until it arrives 
at an admitted absurdity, and, by thus destroying the hypothesis, 
establishes the result originally desired. For it is necessary to under- 
stand generally that all mathematical arguments either proceed from 
the first principles or lead back to them, as Porphyry somewhere says. 
And those which proceed from the first principles are again of two 
kinds, for they start either from common notions and the clearness of 
the self-evident alone, or from results previously proved ; while those 
which lead back to the principles are either by way of assuming the 
principles or by way of destroying them. Those which assume the 
principles are called analyses, and the opposite of these are syntheses — 
for it is possible to start from the said principles and to proceed in 
the regular order to the desired conclusion, and this process is syn- 
thesis — while the arguments which would destroy the principles are 

the first noted in history, it is difficult to suppose that it was really the first ; for Hippocrates 
must have found instances of it in the Pythagorean geometry. Bret Schneider, 1 think, comes 
nearer the truth when he boldly (p. 99) translates: "This reduction <tf tht aforesaid con- 
struct 'iott is said to have been first given by Hippocrates." The words are vpQrroy Si $a<n 
ntr &-ropovtt4vup StaypaftfiAruv r^v iira-ytoy^v ro^iratrffai, which must, literally, be translated 
as in the text above ; but, when Proclus speaks vaguely of " di ffi cult constructions," he 
probably means to say simply that " this first recorded instance of a reduction of a difficult 
construction is attributed to Hippocrates." 

1 Aristotle, Anal, prior. I. 7, 19 b 5 ; 1. 44, 50 a 30. 

" ibid. 1. it, 39 b 35 ; 1. 19, +5 & 35. 

3 Anal. post. I, 14, 85 a 16 etc. * Anal, prior, \. 13, 40 b tj. 

* Anal, prior. I. 53, 41 a 14, • ibid. n. 14, 61 b 19. 

' Proclus, p. 1J4, Ji — 17. 



ch. ix. $6] OTHER TECHNICAL TERMS 137 

called reductiones ad absurdum. For it is the function of this method 
to upset something admitted as clear 1 ." 

8. Analysis and Synthesis. 

It will be seen from the note on Eucl. xin. i that the mss. of the 
Elements contain definitions of Analysis and Synthesis followed by 
alternative proofs of xill. i — 5 aftet that method. The definitions and 
alternative proofs are interpolated, but they have great historical 
interest because of the possibility that they represent an ancient 
method of dealing with these propositions, anterior to Euclid. The 
propositions give properties of a line cut " in extreme and mean ratio," 
and they are preliminary to the construction and comparison of the 
five regular solids. Now Pappus, in the section of his Collection dealing 
with the latter subject", says that he will give the comparisons between 
the five figures, the pyramid, cube, octahedron, dodecahedron and 
icosahedron, which have equal surfaces, " not by means of the so-called 
analytical inquiry, by which some of the ancients worked out the proofs, 
but by the synthetical method 1 ...." The conjecture of Bretschneider 
that the matter interpolated in Eucl. XIII. is a survival of investiga- 
tions due to Eudoxus has at first sight much to commend it 4 . In the 
first place, we are told by Proclus that Eudoxus "greatly added to 
the number of the theorems which Plato originated regarding the 
sectim,&nd employed in them the method of analysis'." It is obvious 
that " the section " was some particular section which by the time of 
Plato had assumed great importance ; and the one section of which 
this can safely be said is that which was called the " golden section," 
namely, the division of a straight line in extreme and mean ratio 
which appears in Eucl. II. 1 1 and is therefore most probably Pytha- 
gorean. Secondly, as Cantor points out 8 , Eudoxus was the founder 
of the theory of proportions in the form in which we find it in Euclid 
v., VI., and it was no doubt through meeting, in the course of his 
investigations, with proportions not expressible by whole numbers 
that he came to realise the necessity for a new theory of proportions 
which should be applicable to incommensurable as well as commen- 
surable magnitudes. The "golden section" would furnish such a case. 
And it is even mentioned by Proclus in this connexion. He is 
explaining' that it is only in arithmetic that all quantities bear 
"rational" ratios (pyros Xoyos) to one another, while in geometry there 
are " irrational " ones (appqros) as well. " Theorems about sections 
like those in Euclid's second Book are common to both [arithmetic 
and geometry] except that in which the straight line is cut in extreme 
and mean ratio"." 

1 Proclus, p, i5j, ' — '** 

s Pappus, v. p. 410 soq, * ibid. pp. +10, 17—411, i. 

* Bretschneider, p. ioB. See however Heiberg's recent suggestion {Paratipomcna zu 
Euklid in Hermts, , XXXVIII., 1W3) that the author was Heron. The suggestion is based 
on a comparison with the remarks on analysis and synthesis quoted from Heron by an-NairtzI 
(ed. Curtie, P- 89) at the beginning of his commentary on Eucl. Book II. On the whole, 
this suggestion commends itself to me more than that of Bretschneider. 

5 Proclus, p. 67, 6. B Cantor, Gtsck. d. Math. I 3 , p. 34 j. 

7 Proclus, p, 60, 7—9. B ibid. p. 60, [6 — 19. 



ij8 INTRODUCTION [ch. ix. }6 

The definitions of Analysis and Synt Justs interpolated in Eucl. 
XIII. are as follows (I adopt the reading of B and V, the only in- 
telligible one, for the second). 

" Analysis is an assumption of that which is sought as if it were 
admitted < and the passage > through its consequences to something 
admitted (to be) true. 

" Synthesis is an assumption of that which is admitted < and the 
passage > through its consequences to the finishing or attainment of 
what is sought." 

The language is by no means clear and has, at the best, to be 
filled out. 

Pappus has a fuller account 1 : 

" The so-called ivakv&fMPW (' Treasury of Analysis ") is, to put it 
shortly, a special body of doctrine provided for the use of those who, 
after finishing the ordinary Elements, are desirous of acquiring the 
power of solving problems which may be set them involving (the 
construction of) lines, and it is useful for this alone. It is the work 
of three men, Euclid the author of the Elements, Apollonius of Perga, 
and Aristaeus the elder, and proceeds by way of analysis and synthesis. 

" Analysis then takes that which is sought as if it were admitted 
and passes from it through its successive consequences to something 
which is admitted as the result of synthesis: for in analysis we assume 
that which is sought as if it were (already) done (yeyovos), and we 
inquire what it is from which this results, and again what is the ante- 
cedent cause of the latter, and so on, until by so retracing our steps 
we come upon something already known or belonging to the class of 
first principles, and such a method we call analysis as being solution 

backwards (deaira\tv \vtrtr). 

" But in synthesis, reversing the process, we take as already done 
that which was last arrived at in the analysis and, by arranging in 
their natural order as consequences what were before antecedents, 
and successively connecting them one with another, we arrive finally 
at the construction of what was sought ; and this we call synthesis. 

" Now analysis is of two kinds, the one directed to searching for 
the truth and called theoretical, the other directed to finding what we 
are told to find and called problematical, (1) In the theoretical kind 
we assume what is sought as if it were existent and true, after which 
we pass through its successive consequences, as if they too were true 
and established by virtue of our hypothesis, to something admitted : 
then (a), if that something admitted is true, that which is sought will 
also be true and the proof will correspond in the reverse order to the 
analysis, but (d), if we come upon something admittedly false, that 
which is sought will also be false. (2) In the problematical kind we 
assume that which is propounded as if it were known, after which we 
pass through its successive consequences, taking them as true, up to 
something admitted : if then (a) what is admitted is possible and 
obtainable, that is, what mathematicians call given, what was originally 
proposed will also be possible, and the proof will again correspond in 

1 Pappus, VII. pp. 63+ — 6. 



ch. ix. $6] OTHER TECHNICAL TERMS 139 

reverse order to the analysis, but if {&) we come upon something 
admittedly impossible, the problem will also be impossible." 

The ancient Analysis has been made the subject of careful studies 
by several writers during the last half-century, the most complete 
being those of Hankel, Duhamel and Zeuthen.; others by Ofterdinger 
and Cantor should also be mentioned'. 

The method is as follows. It is required, let us say, to prove that 
a certain proposition A is true. We assume as a hypothesis that A 
is true and, starting from this we find that, if A is true, a certain 
other proposition B is true ; if B is true, then C ; and so on until 
we arrive at a proposition K which is admittedly true. The object 
of the method is to enable us to infer, in the reverse order, that, since 
K is true, the proposition A originally assumed is true. Now 
Aristotle had already made it clear that false hypotheses might lead 
to a conclusion which is true. There is therefore a possibility of error 
unless a certain precaution is taken. While, for example, B may be a 
necessary consequence of A, it may happen that A is not a necessary 
consequence of B, Thus, in order that the reverse inference from the 
truth of K that A is true may be logically justified, it is necessary 
that each step in the chain of inferences should be unconditionally 
convertible. As a matter of fact, a very large number of theorems in 
elementary geometry are unconditionally convertible, so that in practice 
the difficulty in securing that the successive steps shall be convertible 
is not so great as might be supposed. But care is always necessary. 
For example, as Hankel says 1 , a proposition may not be uncon- 
ditionally convertible in the form in which it is generally quoted. 
Thus the proposition "The vertices of all triangles having a common 
base and constant vertical angle lie on a circle " cannot be converted 
into the proposition that "All triangles with common base and vertices 
lying on a circle have a constant vertical angle"; for this is only true 
if the further conditions are satisfied (]) that the circle passes through 
the extremities of the common base and {2) that only that part of the 
circle is taken as the locus of the vertices which lies on one side of the 
base. If these conditions are added, the proposition is unconditionally 
convertible. Or again, as Zeuthen remarks 3 , K may be obtained by 
a series of inferences in which A or some other proposition in the 
series is only apparently used ; this would be the case e.g. when the 
method of modem algebra is being employed and the expressions on 
each side of the sign of equality have been inadvertently multiplied 
by some composite magnitude which is in reality equal to zero. 

Although the above extract from Pappus does not make it clear 
that each step in the chain of argument must be convertible in the 
case taken, he almost implies this in the second part of the definition 
jf Analysis where, instead of speaking of the consequences B, C... 

1 lAx^.^ZurGtstkichlederMathetnatikinAlterihumMttdMUitiatter^ 1874, pp. 137 — 150; 
Duhamd, Damithodis dans la scititra de raittmnement. Part I., 3 ed., Paris, 1885, pp. 39 — 68 ; 
Zeutben, Gmkuktt der MtUhematik im Attcrtnm und Miltclalttr, 1896, pp. 92 — 104; 
Oflerdinger, Beitrdp zur Gisehkhte der gritchischm MathmnUii, Ulm, i860; Cantor, 
Geschkkte der Mathtmatiki l t , pp. 110 — i. 

1 Hankel, p. 139. * Zeiuhen, p. 103. 



Mo INTRODUCTION [ch. ix. 5 6 

successively following from A, he suddenly changes the expression 
and says that we inquire what it is (B)frem which A follows (A being 
thus the consequence of B, instead of the reverse), and then what 
(viz. C) is the antecedent cause of B; and in practice the Greeks 
secured what was wanted by always insisting on the analysis being 
confirmed by subsequent synthesis, that is, they laboriously worked 
backwards the whole way from K. to A, reversing the order of the 
analysis, which process would undoubtedly bring to light any flaw 
which had crept into the argument through the accidental neglect of 
the necessary precautions. 

Reductio ad absurdum a variety of analysis. 

In the process of analysis starting from the hypothesis that a 
proposition A is true and passing through B, C... as successive con- 
sequences we may arrive at a proposition K which, instead of being 
admittedly true, is either admittedly false or the contradictory of the 
original hypothesis A or of some one or more of the propositions B, C... 
intermediate between A and K. Now correct inference from a true 
proposition cannot lead to a false proposition ; and in this case there- 
fore we may at once conclude, without any inquiry whether the 
various steps in the argument are convertible or not, that the hypo- 
thesis A is false, for, if it were true, all the consequences correctly 
inferred from it would be true and no incompatibility could arise. 
This method of proving that a given hypothesis is false furnishes an 
indirect method of proving that a given hypothesis A is true, since we 
have only to take the contradictory of A and to prove that it is false. 
This is the method of reductio ad absurdum, which is therefore a variety 
of analysis. The contradictory of A, or not-A, will generally include 
more t han one case and, in order to prove its falsity, each of the cases 
must be separately disposed of: e.g., if it is desired to prove that a 
certain part of a figure is equal to some other part, we take separately 
the hypotheses (i) that it is greater, (2) that it is less, and prove 
that each of these hypotheses leads to a conclusion either admittedly 
false or contradictory to the hypothesis itself or to some one of its 
consequences. 

Analysis as applied to problems. 

It is in relation to problems that the ancient analysis has the 
greatest significance, because it was the one general method which 
the Greeks used for solving all "the more abstruse problems" (ra 
dera<f>G(TTepa twv TrpopKufiaTwvy '. 

We have, let us suppose, to construct a figure satisfying a certain 
set of conditions If we are to proceed at all methodically and not 
by mere guesswork, it is first necessary to "analyse" those conditions. 
To enable this to be done we must get them clearly in our minds, 
which is only possible by assuming all the conditions to be actually 
fulfilled, in other words, by supposing the problem solved. Then v;e 
have to transform those conditions, by all the means which practice in 
such cases has taught us to employ, into other conditions which are 
necessarily fulfilled if the original conditions are, and to continue this 

1 Proclus, p. 141, 16, 17. 



CH. ix. $6] OTHER TECHNICAL TERMS 141 

transformation until we at length arrive at conditions which we 
are in a position to satisfy 1 . In other words, we must arrive at 
some relation which enables us to construct a particular part of 
the figure which, it is true, has been hypothetically assumed and 
even drawn, but which nevertheless really requires to be found in 
order that the problem may be solved. From that moment the 
particular part of the figure becomes one of the data, and a fresh 
relation has to be found which enables a fresh part of the figure 
to be determined by means of the original data and the new one 
together. When this is done, the second new part of the figure also 
belongs to the data ; and we proceed in this way until all the parts 
of the required figure are found 1 . The first part of the analysis 
down to the point of discovery of a relation which enables 
us to say that a certain new part of the figure not belonging 
to the original data is given, Hankel calls the transformation ; the 
second part, in which it is proved that all the remaining parts of 
the figure are "given," he calls the resolution. Then follows the 
synthesis, which also consists of two parts, (1) the construction, in 
the order in which it has to be actually carried out, and in general 
following the course of the second part of the analysis, the resolution ; 
(2) the demonstration that the figure obtained does satisfy all the given 
conditions, which follows the steps of the first part of the analysis, 
the transformation, but in the reverse order. The second part of 
the analysis, the resolution, would be much facilitated and shortened 
by the existence of a systematic collection of Data such as Euclid's 
book bearing that title, consisting of propositions proving that, if 
in a figure certain parts or relations me. given, other parts or relations 
are also given. As regards the first part of the analysis, the trans- 
formation, the usual rule applies that every step in the chain must 
be unconditionally convertible ; and any failure to observe this 
condition will be brought to light by the subsequent synthesis. 
The second part, the resolution, can be directly turned into the 
construction since that only is given which can be constructed by 
the means provided in the Elements. 

It would be difficult to find a better illustration of the above than 
the example chosen by Hankel from Pappus*. 

Given a circle ABC and two points D, E external to it, to draw 
straight lines DB, KB from D, E to a point B on the circle such that, 
if DB, KB produced meet the circle again in C, A, AC shall be parallel 
A>DE. 

Analysis. 

Suppose the problem solved and the tangent at A drawn, meeting 
ED produced in F. 

(Part I. Transformation.) 

Then, since AC is parallel to DE, the angle at C is equal to the 
angle CDE. 

But, since FA is a tangent, the angle at C is equal to the angle FAE. 

Therefore the angle FAE is equal to the angle CDE, whence A, 
B, D, Fact concyclic. 

1 Zcutben, p. 93. * Hankel, p. 141. ' Pappus, vn. pp. 630 — 1. 




14* INTRODUCTION [ch. IX. 56 

Therefore the rectangle AE, EB is equal to the rectangle FE, 
ED. 

(Part II, Resolution.) 

But the rectangle AE, EB is given, 
because it is equal to the square on the 
tangent from E. 

Therefore the rectangle FE, ED is 
given ; 

and, since ED is given, FE is given (in 
length). [Data, 57.J 

But FE is given in position also, so 
that F is also given. [Data, 27.] 

Now FA is the tangent from a given point F to a circle A BC 
given in position ; 
therefore FA is given in position and magnitude. [Data, 90.] 

And F is given ; therefore A is given. 

But £ is also given ; therefore the straight line AE is given in 
position, [Data, 26.] 

And the circle ABC is given in position ; 
therefore the point B is also given. [Data, 25.] 

But the points D, E are also given ; 
therefore the straight lines DB, BE are also given in position. 

Synthesis. 

(Part I. Construction) 

Suppose the circle ABC and the points D, E given. 

Take a rectangle contained by ED and by a certain straight 
line EF equal to the square on the tangent to the circle from E. 

From F draw FA touching the circle in A ; join ABE and then 
DB, producing DB to meet the circle at C. Join AC. 

1 say then that A C is parallel to DE. 

(Part II. Demonstration.) 

Since, by hypothesis, the rectangle FE, ED is equal to the square 
on the tangent from E, which again is equal to the rectangle AE, EB, 
the rectangle AE, EB is equal to the rectangle FE, ED, 

Therefore A, B, D, F are concyclic, 
whence the angle FAE is equal to the angle BDE. 

But the angle FAE is equal to the angle ACB in the alternate 
segment ; 
therefore the angle A CB is equal to the angle BDE. 

Therefore AC Is parallel to DEk 

In cases where a &iopt<rft6<; is necessary, i.e. where a solution is 
only possible under certain conditions, the analysis will enable those 
conditions to be ascertained. Sometimes the Stopw/ioe is stated and 
proved at the end of the analysis, e.g. in Archimedes, On the Sphere 
and Cylinder, II. 7 ; sometimes it is stated in that place and the proof 
postponed till after the end of the synthesis, e.g. in the solution of 
the problem subsidiary to Oh the Sphere and Cylinder, II. 4, preserved 
in Eutocius' commentary on that proposition. The analysis should 
also enable us to determine the number of solutions of which the 
problem is susceptible. 



ch. ix. §7] THE DEFINITIONS 143 

I 7. THE DEFINITIONS. 

General. "Real" and "Nominal" Definitions. 

It is necessary, says Aristotlej whenever any one treats of any 
whole subject, to divide the genus into its primary constituents, those 
which are indivisible in species respectively : e.g. number must be 
divided into triad and dyad ; then an attempt must be made in this 
way to obtain definitions, e.g. of a straight line, of a circle, and of 
a right angle 1 . 

The word for definition is 3po?. The original meaning of this 
word seems to have been " boundary," " landmark." Then we have 
it in Plato and Aristotle in the sense of standard or determining 
principle ("id quo alicuius rei natura constituitur vel definitur," 
Index Aristotelicus) 1 ; and closely connected with this is the sense of 
definition. Aristotle uses both JSpo? and 6pnrfia<t for definition, the 
former occurring more frequently in the Topics, the latter in the 
Metaphysics. 

Let us now first be clear as to what a definition does not do. 
There is nothing in connexion with definitions which Aristotle takes 
more pains to emphasise than that a definition asserts nothing as to 
the existence or non-existence of the thing defined. It is an answer 
to the question what a thing is (ri 4cm), and does not say that it 
is (5rt cart). The existence of the various things defined has to be 
proved, except in the case of a few primary things in each science, 
the existence of which is indemonstrable and must be assumed among 
the first principles of each science ; e.g. points and lines in geometry 
must be assumed to exist, but the existence of everything else must 
be proved This is stated clearly in the long passage quoted above 
under First Principles'. It is reasserted in such passages as the 
following. " The (answer to the question) what is a man and the 
fact that a man exists are different things'." " It is clear that, even 
according to the view of definitions now current, those who define 
things do not prove that they exist 1 ." " We say that it is by 
demonstration that we must show that everything exists, except 
essence (tl fii) ova-la eiij). But the existence of a thing is never 
essence; for the existent is not a genus. Therefore there must be 
demonstration that a thing exists. Thus, what is meant by triangle 
the geometer assumes, bat that it exists he has to prove 11 ." "Anterior 
knowledge of two sorts is necessary : for it is necessary to presuppose, 
with regard to some things, that they exist; in other cases it is 
necessary to understand what the thing described is, and in other 
cases it is necessary to do both. Thus, with the fact that one of two 
contradictories must be true, we must know that it exists (is true); 

1 Anal, pest, tt. 13, 96 b ij. 

' Of. Dt anitna, I. 1, 404 a g, where " breathing" is spoken of as the Spot of " life," and 
the many passages in the Politic! where the wurd is used to denote that which gives its 
special character to the several forms of government (virtue being the &poi of aristocracy, 
wealth of oligarchy, liberty of democracy, 130.4 a 10) ', Plato, fapuhtic, vju, 551 c 

* Anal. post. I. 10, 76 a 31 sqq. * ibid. IJ. 7, 91 b 10* 

* ibid. 91 b 19. ' ibid. 91 h 11 sqq. 



144 INTRODUCTION [at. ix. § 7 

of the triangle we must know that it means such and such a thing ; of 
the unit we must know both what it means and that it exists 1 ." What 
is here so much insisted on is the very fact which Mill pointed out 
in his discussion of earlier views of Definitions, where he says that 
the so-called real definitions or definitions of things do not constitute 
a different kind of definition from nominal definitions, or definitions 
of names ; the former is simply the latter plus something else, namely 
a covert assertion that the thing defined exists, "This covert assertion 
is not a definition but a postulate. The definition is a mere identical 
proposition which gives information only about the use of language, 
and from which no conclusion affecting matters of fact can possibly 
be drawn. The accompanying postulate, on the other hand, affirms 
a fact which may lead to consequences of every degree of importance. 
It affirms the actual or possible existence of Things possessing the 
combination of attributes set forth in the definition : and this, if true, 
may be foundation sufficient on which to build a whole fabric of 
scientific truth 1 ." This statement really adds nothing to Aristotle's 
doctrine' ; it has even the slight disadvantage, due to the use of 
the word "postulate" to describe "the covert assertion" in all cases, 
of not definitely pointing out that there are cases where existence 
has to be proved as distinct from those where it must be assumed. 
It is true that the existence of a definiend may have to be taken 
for granted provisionally until the time comes for proving it; but, 
so far as regards any case where existence must be proved sooner 
or later, the provisional assumption would be for Aristotle, not a 
postulate, but a hypothesis. In modern times, too, Mill's account of 
the true distinction between real and nominal definitions had been 
fully anticipated by Saccheri 1 , the editor of Euclides ab omni naevo 
vindicatas (1733), famous in the history of non-Euclidean geometry. 
In his Logica Demonstrativa (to which he also refers in his Euclid) 
Saccheri lays down the clear distinction between what he calls de- 
finitiones quid nominis or nominales, and definitiones quid rei or reales, 
namely that the former are only intended to explain the meaning 

1 Anal. post. I. i, 71 a II sqq. s Mill's Syiltm of Logic, Bk. [. ch. viii. 

* It istrup ihat it was in opposition to *Mhe ideas of most of the Aristoteiian logicians" 
(rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his 
observation : " We have already made, and shall often have to repeat, the remark, that the 

Ehilosophers who overthrew Realism by no means got rid of the consequences of Realism, 
ut retained long afterwards, in their own philosophy, numerous propositions which could 
only have a rational meaning as part of a Realistic system. It had been handed down from 
Anstotle, and probably from earlier times, as an obvious truth, that the science of geometry 
is deduced from definitions. This, so long as a definition was considered to be a proposition 
' unfolding the nature of the thing,' did well enough. But Hobbes followed and rejected 
utterly the notion that a definition declares the nature of the thing, or does anything but 
state the meaning of a name ; yet he continued to affirm as broadly as any of his predecessors 
that the dpx&l, prituipia, or original premisses of mathematics, and even of all science, are 
definitions ; producing the singular paradox that systems of scientific truth, nay, all truths 
whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of 
mankind concerning the signification of words." Aristotle was guilty of no such paradox ; 
on the contrary, he exposed it as plainly as did Mill. 

* This has been fully brought out in two papers by G. Vailati, La ttaria Ariitottiita del/a 
definiziow [Rivista di Filosofia s sciertse affini. 190;}, and Di un l opera dimtrUicata dtl 
P. Gsrolanto Satthtri (*' Logica Demonstrativa," 1097) (in Hivista Fifosefica, 1903). 



en. IX. 5 7] THE DEFINITIONS 145 

that is to be attached to a given term, whereas the latter, besides 
declaring the meaning of a word, affirm at the same time the existence 
of the thing defined or, in geometry, the possibility of constructing it. 
The definitio quid nominis becomes a definitio quid rei " by means of a 
postulate, or when we come to the question whether the thing exists and 
it is answered affirmatively 1 ." Definitiones quid nominis are in them- 
selves quite arbitrary, and neither require nor are capable of proof; 
they are merely provisional and are only intended to be turned as 
quickly as possible into definitiones quid rei, either (1) by means of 
a postulate in which it is asserted or conceded that what is defined 
exists or can be constructed, e.g. in the case of straight lines and 
circles, to which Euclid's first three postulates refer, or (2) by 
means of a demonstration reducing the construction of the figure 
defined to the successive carrying-out of a certain number of those 
elementary constructions, the possibility of which is postulated. Thus 
definitiones quid rei are in general obtained as the result of a series of 
demonstrations. Saccheri gives as an instance the construction of a 
square in Euclid I. 46. Suppose that it is objected that Euclid had 
no right to define a square, as he does at the beginning of the Book, 
when it was not certain that such a figure exists in nature; the 
objection, he says, could only have force if, before proving and making 
the construction, Euclid had assumed the aforesaid figure as given. 
That Euclid is not guilty of this error is clear from the fact that 
he never presupposes the existence of the square as defined until 
after I. 46. 

Confusion between the nominal and the real definition as thus de- 
scribed, i.e. the use of the former in demonstration before it has been 
turned into the latter by the necessary proof that the thing defined 
exists, is according to Saccheri one of the most fruitful sources of 
illusory demonstration, and the danger is greater in proportion to 
the "complexity" of the definition, i.e. the number and variety of 
the attributes belonging to the thing defined. For the greater is the 
possibility that there may be among the attributes some that are 
incompatible, i.e. the simultaneous presence of which in a given figure 
can be proved, by means of other postulates etc. forming part of the 
basis of the science, to be impossible. 

The same thought is expressed by Leibniz also. " If," he says, 
" we give any definition, and it is not clear from it that the idea, which 
we ascribe to the thing, is possible, we cannot rely upon the demon- 
strations which we have derived from that definition, because, if that 
idea by chance involves a contradiction, it is possible that even con- 
tradictories may be true of it at one and the same time, and thus our 
demonstrations will be useless. Whence it is clear that definitions 
are not arbitrary. And this is a secret which is hardly sufficiently 
known'." Leibniz' favourite illustration was the " regular polyhedron 
with ten faces," the impossibility of which is not obvious at first sight. 

1 "' Definitio quid neminit nata est evadere definitio quid rri per poiitttaium vel dum 
venitur ad quaestionem an at et respondetur affirmative. ' 

' Opusottts ttfragmenUinidUtdiLtibmi, Paris, Mean, 1903, p. 4 ji. Quoted by Vailati. 



146 INTRODUCTION [ch. ix. { 7 

It need hardly be added that, speaking generally, Euclid's defini- 
tions, and his use of them, agree with the doctrine of Aristotle 
that the definitions themselves say nothing as to the existence of the 
things defined, but that the existence of each of them must be 
proved or (in the case of the " principles ") assumed. In geometry, 
says Aristotle, the existence of points and lines only must be as- 
sumed, the existence of the rest being proved. Accordingly Euclid's 
first three postulates declare the possibility of constructing straight 
lines and circles (the only " lines " except straight lines used in the 
Elements). Other things are defined and afterwards constructed and 
proved to exist : e.g. in Book I., Def. 20, it is explained what is meant 
by an equilateral triangle ; then (1. 1 ) it is proposed to construct it, 
and, when constructed, it is proved to agree with the definition. 
When a square is defined (1. Def. 22), the question whether such a 
thing really exists is left open until, in I. 46, it is proposed to construct 
it and, when constructed, it is proved to satisfy the definition 1 . 
Similarly with the right angle (I. Def. 10, and I. 11) and parallels 
(I. Def. 23, and I. 27—29). The greatest care is taken to exclude 
mere presumption and imagination. The transition from the sub- 
jective definition of names to the objective definition of things is 
made, in geometry, by means of constructions (the first principles of 
which are postulated), as in other sciences it is made by means of 
experience 1 . 

Aristotle's requirements in a definition. 

We now come to the positive characteristics by which, according 
to Aristotle, scientific definitions must be marked. 

First, the different attributes in a definition, when taken separately, 
cover more than the notion defined, but the combination of them 
does not Aristotle illustrates this by the " triad," into which enter 
the several notions of number, odd and prime, and the last " in both 
its two senses (a) of not being measured by any (other) number (<as 
firj fiiTpdadai dpiBfUii) and (£) of not being obtainable by adding 
numbers together " (<uv pA auytceiaBat, e£ aptQfi&v), a unit not being a 
number. Of these attributes some are present in all other odd 
numbers as well, while the last [primeness in the second sense] 
belongs also to the dyad, but in nothing but the triad are they all 
present'." The fact can be equally well illustrated from geometry. 
Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter 
the several notions of figure, four-sided, equilateral, and right-angled, 
each of which covers more than the notion into which all enter as 
attributes 4 . 

Secondly, a definition must be expressed in terms of things which 
are prior to, and better known than, the things defined'. This is 

1 Trendelenburg, Elements Logites Aristotikot, % 50, 

' Trendelenburg, ErlSuttrungcn tu den Elementcn dtr oristotdischen Logii, 3 ed. p. 107. 
On construction as proof of existence in ancient geometry cf. H. G- Zeuthen, Dit gcomctrische 
Construction ctls * Existtns&euieis il in drr antiktn Geometric (in McUhcmatischt Annaien, 
4}. Band). 

* Anal. pest. II. 13, 96 a 33 — b t. 

' Trendelenburg, Erlauttrnngtn, p. 108. ' Topics vr. 4, 141116 sqq. 



ch. IX, $ 7] THE DEFINITIONS 147 

clear, since the object of a definition is to give us knowledge of the 
thing defined, and it is by means of things prior and better known 
that we acquire fresh knowledge, as in the course of demonstrations. 
But the terms " prior " and " better known " are, as usual susceptible 
of two meanings ; they may mean (1 ) absolutely or logically prior and 
better known, or (2) better known relatively to us. In the absolute 
sense, or from the standpoint of reason, a point is better known than 
a line, a line than a plane, and a plane than a solid, as also a unit is 
better known than number (for the unit is prior to, and the first 
principle of, any number). Similarly, in the absolute sense, a letter is 
prior to a syllable. But the case is sometimes different relatively to 
us ; for example, a solid is more easily realised by the senses than a 
plane, a plane than a line, and a line than a point. Hence, while it is 
more scientific to begin with the absolutely prior, it may, perhaps, be 
permissible, in case the learner is not capable of following the scientific 
order, to explain things by means of what is more intelligible to him. 
"Among the definitions framed on this principle are those of the 
point, the line and the plane ; all these explain what is prior by 
means of what is posterior, for the point is described as the extremity 
of a line, the line of a plane, the plane of a solid." But, if it is asserted 
that such definitions by means of things which are more intelligible 
relatively only to a particular individual are really definitions, it will 
follow that there may be many definitions of the same thing, one for 
each individual for whom a thing is being defined, and even different 
definitions for one and the same individual at different times, since at 
first sensible objects are more intelligible, while to a better trained 
mind they become less so. It follows therefore that a thing should 
be defined by means of the absolutely prior and not the relatively 
prior, in order that there may be one sole and immutable definition. 
This is further enforced by reference to the requirement that a good 
definition must state the genus and the differentiae, for these are 
among the things which are, in the absolute sense, better known than, 
and prior to, the species (twv AttX&s yvwptfMoTipwv teal irpor^pav tov 
etBovs tffrtv). For to destroy the genus and the differentia .is to 
destroy the species, so that the former are prior to the species ; they 
are also better known, for, when the species is known, the genus and 
the differentia must necessarily be known also, e.g. he who knows 
"man" must also know "animal" and "land-animal," but it does not 
follow, when the genus and differentia are known, that the species is 
known too, and hence the species is Jess known than they are 1 . It 
may be frankly admitted that the scientific definition will require 
superior mental powers for its apprehension ; and the extent of its 
use must be a matter of discretion. So far Aristotle ; and we have 
here the best possible explanation why Euclid supplemented his 
definition of a point by the statement in 1. Def, 3 that the extremities of 
a line are points and his definition of a surface by I. Def. 6 to the effect 
that the extremities of a surface are lines. The supplementary expla- 

1 'Tafia vi. 4, C41 b 35 — 34, 



148 INTRODUCTION [ch. ix. } 7 

nations do in fact enable us to arrive at a better understanding of the 
formal definitions of a point and a line respectively, as is well ex- 
plained by Simson in his note on Def. I, Simson says, namely, that 
we must consider a solid, that is, a magnitude which has length, 
breadth and thickness, in order to understand aright the definitions of 
a point, a line and a surface. Consider, for instance, the boundary 
common to two solids which are contiguous or the boundary which 
divides one solid into two contiguous parts; this boundary is a surface. 
We can prove that it has no thickness by taking away either solid, 
when it remains the boundary of the other; for, if it had thickness, the 
thickness must either be a part of one solid or of the other, in which 
case to take away one or other solid would take away the thickness 
and therefore the boundary itself: which is impossible. Therefore 
the boundary or the surface has no thickness. In exactly the same 
way, regarding a line as the boundary of two contiguous surfaces, we 
prove that the line has no breadth ; and, lastly, regarding a point as 
the common boundary or extremity of two lines, we prove that a 
point has no length, breadth or thickness. 

Aristotle on unscientific definitions. 

Aristotle distinguishes three kinds of definition which are un- 
scientific because founded on what is not prior (jtfy *W -n-poTipav). The 
first is a definition of a thing by means of its opposite, e,g. of " good " 
by means of " bad " ; this is wrong because opposites are naturally 
evolved together, and the knowledge of opposites is not uncommonly 
regarded as one and the same, so that one of the two opposites 
cannot be better known than the other. It is true that, in some 
cases of opposites, it would appear that no other sort of definition is 
possible: e.g. it would seem impossible to define double apart from the 
half and, generally, this would be the case with things which in their 
very nature {/cad* aura) are relative terms (w/109 Tt \eyerai), since one 
cannot be known without the other, so that in the notion of either the 
other must be comprised as well 1 . The second kind of definition 
which is based on what is not prior is that in which there is a 
complete circle through the unconscious use in the definition itself of 
the notion to be defined though not of the name*. Trendelenburg 
illustrates this by two current definitions, (i) that of magnitude as 
that which can be increased or diminished, which is bad because the 
positive and negative comparatives " more " and " less " presuppose 
the notion of the positive " great," (2) the famous Euclidean definition 
of a straight line as that which " lies evenly with the points on itself" 
(££ Xaov to*s e<f>' lavTrjii <rij^eiois Keirai), where "lies evenly" can only 
be understood with the aid of the very notion of a straight line which is 
to be defined'. The third kind of vicious definition from that which 
is not prior is the definition of one of two coordinate species by means 
of its coordinate (dirtiStypiiftivov), e.g. a definition of " odd " as that 
which exceeds the even by a unit (the second alternative in Eucl. VII. 
Def. 7) ; for " odd " and " even " are coordinates, being differentiae of 

1 Topki vi. 4, 141 a u — 31. * Hid, 141 a 34— b 6. 

* Trendelenburg, Erlautentngtn, p. iij. 



ch. IX. J 7] THE DEFINITIONS 149 

number'. This third kind is similar to the first. Thus, says Tren- 
delenburg, it would be wrong to define a square as "a rectangle 
with equal sides." 

Aristotle's third requirement. 

A third general observation of Aristotle which is specially relevant 
to geometrical definitions is that "to know what a thing is (ri itrTtv) is 
the same as knowing why it is (foil ri ia-nv)'." " What is an eclipse ? 
A deprivation of light from the moon through the interposition of the 
earth. Why does an eclipse take place? Or why is the moon 
eclipsed ? Because the light fails through the earth obstructing it 
What is harmony ? A ratio of numbers in high or low pitch. Why 
does the high-pitched harmonise with the low-pitched? Because 
the high and the low have a numerical ratio to one another*." " We 
seek the cause (ri Sto-rt) when we are already in possession of the 
/act (rd ot(). Sometimes they both become evident at the same time, 
but at al! events the cause cannot possibly be known [as a cause] 
before the fact is known*." " It is impossible to know what a thing is 
if we do not know that it is'" Trendelenburg paraphrases: "The 
definition of the notion does not fulfil its purpose until it is made 
genetic. It is the producing cause which first reveals the essence of 

the thing The nominal definitions of geometry have only a 

provisional significance and are superseded as soon as they are made 
genetic by means of construction." E.g. the genetic definition of a 
parallelogram is evolved from Eucl. I. 3 1 (giving the construction for 
parallels) and I, 33 about the lines joining corresponding ends of two 
straight lines parallel and equal in length. Where existence is proved 
by construction, the cause and the fact appear together 1 . 

Again, " it is not enough that the defining statement should set 
forth the fact, as most definitions do; it should also contain and 
present the cause ; whereas in practice what is stated in the definition 
is usually no more than a conclusion (o-vfiiripatrfta). For example, 
what is quadrature ? The construction of an equilateral right-angled 
figure equal to an oblong. But such a definition expresses merely the 
conclusion. Whereas, if you say that quadrature is the discovery of a 
mean proportional, then you state the reason'." This is better under- 
stood if we compare the statement elsewhere that " the cause is the 
middle term, and this is what is sought in all cases'," and the illustra- 
tion of this by the case of the proposition that the angle in a semi- 
circle is a right angle. Here the middle term which it is sought to 
establish by means of the figure is that the angle in the semi-circle is 
equal to the half of two right angles. We have then the syllogism : 
Whatever is half of two right angles is a right angle ; the angle in a 
semi-circle is the half of two right angles ; therefore {conclusion) the 
angle in a semi-circle is a right angle*. As with the demonstration, so 



1 Topiavi, +, 141 b 7—10. > Anal. post. II. 1, 90 a 31. 

' Anal. post. II. », 90 & 1} — 11. * ibid. u. 8, 93 * 17. 

* ibid, 93 a 10. ' Trendelenburg, Erlduttrungtn, p. no. 

* Di amma II, 1, 413 a 13— JO. * Anal. posU II. a, 00 a 6, 
•■Mttvu, 94 a 18. 



ISO INTRODUCTION [ch. ix. §7 

it should be with the definition, A definition which is to show the 
genesis of the thing defined should contain the middle term or cause ; 
otherwise it is a mere statement of a conclusion. Consider, for 
instance, the definition of " quadrature " as " making a square equal in 
area to a rectangle with unequal sides," This gives no hint as to 
whether a solution of the problem is possible or how it is solved : but, 
if you add that to find the mean proportional between two given 
straight lines gives another straight line such that the square on it is 
equal to the rectangle contained by the first two straight lines, you 
supply the necessary middle term or cause 1 . 

Technical term a not defined by Euclid. 

It will be observed that what is here defined, " quadrature " or 
" squaring " (TeTpaiyawio-^oc), is not a geometrical figure, or an attribute 
of such a figure or a part of a figure, but a technical term used to 
describe a certain problem. Euclid does not define such things ; but 
the fact that Aristotle alludes to this particular definition as well as to 
definitions of deflection (tce/ckd-a-Oat) and of verging {vevew) seems to 
show that earlier text-books included among definitions explanations 
of a number of technical terms, and that Euclid deliberately omitted 
these explanations from his Elements as surplusage. Later the 
tendency was again in the opposite direction, as we see from the much 
expanded Definitions of Heron, which, for example, actually include 
a definition of a deflected line (KettXaajUw) ypa/i,^)'. Euclid uses the 
passive of icTJiv occasionally*, but evidently considered it unnecessary 
to explain such terms, which had come to bear a recognised meaning. 

The mention too by Aristotle of a definition of verging (vcveiv) 
suggests that the problems indicated by this term were not excluded 
from elementary text-books before Euclid. The type of problem 
(vev<ri<t) was that of placing a straight line across two lines, e.g. two 
straight lines, or a straight line and a circle, so that it shall verge to a 
given point (i.e. pass through -it if produced) and at the same time the 
intercept on it made by the two given lines shall be of given length. 

1 Other passages in Aristotle may be quoted to the tike effect: e.g.- Anal. pest. I. i, 
71 b 9 " We consider that we know a particular thing in the absolute sense, as distinct 
from the sophistical and incidental sense, when we consider that we know the cause on 
account of which the thing is, in the sense of knowing that it is the cause of that thing and 
that it cannot be otherwise," ibid. I. 15, 79 a 1 " For here to know the fact is the function of 
those who are concerned with sensible things, to know the cause [s the function of the mathe* 
matician ; it is he who possesses the proofs of the causes, and often he does not know the 
fact." In view of such passages it is difficult to see how Proclus came to write (p. 101, i i) 
that Aristotle was the originator ( ' Apiaror&ovt Kardp^curroi) of the idea of Ampbtnomus and 
others that geometry does not investigate the cause and the why (ri 9i& ri}. To this Gerninuj 
replied that the investigation of the cause does, on the contrary, appear in geometry. " For 
how can it be maintained that it is not the business of the geometer to inquire for what reason, 
on the one hand, an infinite number of equilateral polygons are inscribed in a circle, but, on 
the other hand, it is not possible to inscribe in a sphere an infinite number of polyhedral 
figures, equilateral, equiangular, and made up of similar plane figures ? Whose business is it 
to ask this question and find the answer to it if it is not that of the geometer? Now when 
geometers reason per impoisibilt they are content to discover the property, bat when they 
argue by direct proof, if such proof be only partial {irl liipovt), this does not suffice for 
showing the cause ; if however it is general and applies to all like cases, the why (ri fed ri) 
is at once and concurrently made evident." 

1 Heron, Def. 11 (vol.* iv, Heib. pp. 11-14). * e.g. in lit. jo and in Data 89, 



ch. ix. 57] THE DEFINITIONS 151 

In genera!, the use of conies is required for the theoretical solution of 
these problems, or a mechanical contrivance for their practical 
solution 1 . Zeuthen, following Oppermann, gives reasons for supposing, 
not only that mechanical constructions were practically used by the 
older Greek geometers for solving these problems, but that they were 
theoretically recognised as a permissible means of solution when the 
solution could not be effected by means of the straight line and circle, 
and that it was only in later times that it was considered necessary to 
use conies in every case where that was possible*. Heiberg' suggests 
that the allusion of Aristotle to vtvo-em perhaps confirms this sup- 
position, as Aristotle nowhere shows the slightest acquaintance with 
conies. I doubt whether this is a safe inference, since the problems 
of this type included in the elementary text-books might easily have 
been limited to those which could be solved by " plane " methods (i.e. 
by means of the straight line and circle). We know, e.g., from Pappus 
that Apollonius wrote two Books on plane rewetf*. But one thing 
is certain, namely that Euclid deliberately excluded this class of 
problem, doubtless as not being essential in a book of Elements. 

Definitions not afterwards used. 

I-astly, Euclid has definitions of some terms which he never after- 
wards uses, e.g. oblong (eTepo/w^s), rhombus, rhomboid. The "oblong" 
occurs in Aristotle ; and it is certain that all these definitions are 
survivals from earlier books of Elements. 

1 Cf. the chapter on *c6aen in The Works of Archimedes, pp. c — exxii. 

* Zeutherii Die Lekrt von dm Kegtlsihnittcn im Alttrtum, ch. is, p. 16a. 

* Heiberg, Matkemaiisches tu A rh tot cits, p. 16. 
1 Pappiu VH. pp. 670—1. 



BOOK I. 

DEFINITIONS. 

i. A point is that which has no part. 

2. A line is bread thless length, 

3. The extremities of a line are points. 

4. A straight line is a line which lies evenly with the 
points on itself. 

5. A surface is that which has length and breadth only. 

6. The extremities of a surface are lines. 

7. A plane surface is a surface which lies evenly with 
the straight lines on itself. 

8. A plane angle is the inclination to one another of 
two lines in a plane which meet one another and do not lie in 
a straight line. 

9. And when the lines containing the angle are straight, 
the angle is called rectilineal. 

10. When a straight line set up on a straight line makes 
the adjacent angles equal to one another, each of the equal 
angles is right, and the straight line standing on the other is 
called a perpendicular to that on which it stands. 

n. An obtuse angle is an angle greater than a right 
angle. 

12. An acute angle is an angle less than a right angle. 

13. A boundary is that which is an extremity of any- 
thing. 

14. A figure is that which is contained by any boundary 
or boundaries. 

15. A circle is a plane figure contained by one line such 
that all the straight lines falling upon it from one point among 
those lying within the figure are equal to one another ; 



154 BOOK I [i. def. 16— post. 4 

16. And the point is called the centre of the circle. 

17. A diameter of the circle is any straight line drawn 
through the centre and terminated in both directions by the 
circumference of the circle, and such a straight line also 
bisects the circle. 

18. A semicircle is the figure contained by the diameter 
and the circumference cut off by it And the centre of the 
semicircle is the same as that of the circle. 

19. Rectilineal figures are those which are contained 
by straight lines, trilateral figures being those contained by 
three, quadrilateral those contained by four, and multi- 
lateral those contained by more than four straight lines. 

20. Of trilateral figures, an equilateral triangle is that 
which has its three sides equal, an isosceles triangle that 
which has two of its sides alone equal, and a scalene 
triangle that which has its three sides unequal. 

21. Further, of trilateral figures, a right-angled tri- 
angle is that which has a right angle, an obtuse-angled 
triangle that which has an obtuse angle, and an acute - 
angled triangle that which has its three angles acute. 

22. Of quadrilateral figures, a square is that which is 
both equilateral and right-angled ; an oblong that which is 
right-angled but not equilateral ; a rhombus that which is 
equilateral but not right-angled ; and a rhomboid that which 
has its opposite sides and angles equal to one another but is 
neither equilateral nor right-angled. And let quadrilaterals 
other than these be called trapezia. 

23. Parallel straight lines are straight lines which, 
being in the same plane and being produced indefinitely in 
both directions, do not meet one another in either direction. 

POSTULATES. 

Let the following be postulated : 

1. To draw a straight line from any point to any point. 

2. To produce a finite straight line continuously in a 
straight line. 

3. To describe a circle with any centre and distance. 

4. That all right angles are equal to one another. 



i. post. 5— c. n. s] DEFINITIONS ETC. 155 

5. That, if a straight line falling on two straight lines 
make the interior angles on the same side less than two right 
angles, the two straight lines, if produced indefinitely, meet 
on that side on which are the angles less than the two right 
angles. 

COMMON NOTIONS. 

1. Things which are equal to the same thing are also 
equal to one another. 

2. If equals be added to equals, the wholes are equal. 

3. If equals be subtracted from equals, the remainders 
are equal. 

[7] 4> Things which coincide with one another are equal to 
one another. 

[8] 5. The whole is greater than the part 

Definition i. 

Si)/i ttov ia-Ttv, uv [icpiK oi$iy. 

A point is that which has no pari. 

An exactly parallel use of fiepos (itrrt) in the singular is found in Aristotle, 
Metaph. 1035 b 32 fitpos /11F ow tort (tat tqv mSovi, literally "There is a 
part even of the form "; Borsitz translates as if the plural were used, "Theile 
giebt es," and the meaning is simply "even the form is divisible (into parts)." 
Accordingly it would be quite justifiable to translate in this case "A point is 
that which is indivisible into parts." 

Martianus Capella (5th c. a.d.) alone or almost alone translated differently, 
"Punctum est cuius pars nihil est," "a point is that a part of which is netting." 
Notwithstanding that Max Simon (Euclid vnd die sechs planimetrischen Sucker, 
1 901) has adopted this translation (on grounds which I shall presently mention), 
I cannot think that it gives any sense. If a part of a point is nothing, Euclid 
might as well have said that a point is itself "nothing," which of course he 
does not do. 

Pre -Euclidean definitions. 

It would appear that this was not the definition given in earlier text- 
books; for Aristotle (Topics vi. 4, 141 b 20), in speaking of "the definitions" 
of point, line, and surface, says that they alt define the prior by means of the 
posterior, a point as an extremity of a line, a line of a surface, and a surface 
of a solid 

The first definition of a point of which we hear is that given by the 
Pythagoreans (cf. Proclus, p. 95, 21), who defined it as a "monad having 
position" or "with position added" (m°vo; irpoo-kafjowra 9i<rw). It is frequently 
used by Aristotle, either in this exact form (cf. De anima 1. 4, 409 a 6) or its 
equivalent: e.g. in Metaph. 1016 b 24 he says that that which is indivisible 
every way in respect of magnitude and qu& magnitude but has not position is 
a monad, while that which is similarly indivisible and has position is a point. 

Plato appears to have objected to this definition. Aristotle says (Metaph. 



IS* BOOK I [i. dkf. i 

992 a 20) that he objected "to this genus [that of points] as being a geometrical 
fiction (ytinptTpiKav Soyita), and called a point the beginning of a line {&px*l 
ypaft/),^), while again he frequently spoke of * indivisible lines. 1 " To which 
Aristotle replies that even " indivisible lines " must have extremities, so that 
the same argument which proves the existence of lines can be used to prove 
that points exist It would appear therefore that, when Aristotle objects to 
the definition of a point as the extremity of a line (wipe.? ypap.p.ys) as un- 
scientific {Topics vi. 4, 141 b 21), he is aiming at Plato. Heiberg conjectures 
(Mathematisthes m Aristoteks, p. 8) that it was due to Plato's influence that 
the word for "point" generally used by Aristotle {<my/iij) was replaced by 
<n)iitiov (the regular term used by Euclid, Archimedes and later writers), the 
latter term (-nota, a conventional mark) probably being considered more 
suitable than trrtypy (a. puncture) which might appear to claim greater reality 
for a point 

Aristotle's conception of a point as that which is indivisible and has 
position is further illustrated by such observations as that a point is not a 
body (l)e caeh 11. 13, 196 a 17) and has no weight (ibid. 111. 1, 299 a 30); 
again, we can make no distinction between a point and the place (jorm) where 
it is (Physics iv. 1, 209 a n). He finds the usual difficulty in accounting for 
the transition from the indivisible, or infinitely small, to the finite or divisible 
magnitude. A point being indivisible, no accumulation of points, however far 
it may be carried, can give us anything divisible, whereas of course a line is a 
divisible magnitude. Hence he holds that points cannot make up anything 
continuous like a line, point cannot be continuous with point (06 yap l<rrar 
t^ofitvov tnjpMov trtjfitiov y ariy firj tniy/ofi, De gen. el corr. I. 2, 317 a ro), and 
a line is not made up of points (ou ^vy«€i™t in emy/uly, Physics iv, 8, a 1 5 
b 19). A point, he says, is like the now in time: now is indivisible and is 
not a. part of time, it b only the beginning or end, or a division, of time, and 
similarly a point may be an extremity, beginning or division of a line, but is 
not part of it or of magnitude (cf. De eaelo m. 1, 300 a 14, Physics IV. n, 
220 a 1 — 21, vi. 1, 231 b 6 sqq.). It is only by motion that a point can 
generate a line {De anima 1. 4, 409 a 4) and thus be the origin of magnitude. 

Other ancient definitions. 

According to an-Nairizi (ed. Curtee, p. 3) one "Herundes" (not so far 
identified) defined a point as " the indivisible beginning of all magnitudes," 
and Position ius as "an extremity which has no dimension, or an extremity of 
a line." 

Criticisms by commentators. 

Euclid's definition itself is of course practically the same as that which 
Aristotle's frequent allusions show to have been then current, except that it 
omits to say that the point must have position. Is it then sufficient, seeing 
that there are other things which are without parts or indivisible, e.g. the now 
in time, and the unit in number f Proclus answers {p. 93, 18) that the point 
is the only thing in the subject-matter of geometry that is indivisible. Relatively 
therefore to the particular science the definition is sufficient Secondly, the 
definition has been over and over again criticised because it is purely negative. 
Proclus' answer to this is (p. 94, 10) that negative descriptions are appropriate 
to first principles, and he quotes Pa mien ides as having described his first and 
last cause by means of negations merely. Aristotle too admits that it may 
sometimes be necessary for one framing a definition to use negations, e.g. in 
defining privative terms such as "blind"; and he seems to accept as proper 



I okf. i] NOTE ON DEFINITION r 157 

the negative element in the definition of a point, since he says (De anima 111,6, 
430 b 20) that "the point and every division [e.g. in a length or in a period 
of time], and that which is indivisible in this sense, ts exhibited as privation 

Simplicius (quoted by an-Nairizi) says that " a point is the beginning of 
magnitudes and that from which they grow ; it is also the only thing which, 
having position, is not divisible." He, like Aristotle, adds that it is by its 
motion that a point can generate a magnitude : the particular magnitude can 
only be "of one dimension," viz. a line, since the point does not "spread 
itself" (dimittat), Simplicius further observes that Euclid defined a point 
negatively because it was arrived at by detaching surface from body, line from 
surface, and finally point from line. "Since then body has three dimensions 
it follows that a point [arrived at after successively eliminating all three 
dimensions] has none of the dimensions, and has no part," This of course 
reappears in modern treatises (cf. Rausenberger, Eiementar'gtomctrit des 
Punktes, der Geraden und der Ebene, 1887, p. 7). 

An-Nairizi adds an interesting observation. " If any one seeks to know 
the essence of a point, a thing more simple than a line, let him, in the sensible 
world, think of the centre of the universe and the poles." But there is 
nothing new under the sun : the same idea is mentioned, in an Aristotelian 
treatise, in controverting those who imagine that the poles have some influence 
in the motion of the sphere, "when the poles have no magnitude but are 
extremities and points " (De motu animalium 3, 699a 11). 

Modern views. 

In the new geometry represented by the excellent treatises which start 
from new systems of postulates or axioms, the result of the profound study of 
the fundamental principles of geometry during recent years (I need only 
mention the names of Pasch, Veronese, Enriques and Hilbert), points come 
before lines, but the vain effort to define them a priori is not made ; instead 
of this, the nearest material things in nature are mentioned as illustrations, 
with the remark that it is from them that we can get the abstract idea. Cf. 
the full statement as regards the notion of a point in Weber and Wellstein, 
Encyehpddie dtr eiementaren Mathematik, 11., 1905, p. 9. "This notion is 
evolved from the notion of tbe real or supposed material point by the process 
of limits, i.e. by an act of the mind which sets a term to a series of presen- 
tations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam, 
which continually becomes smaller and smaller. In this way vanishes more 
and more the possibility of determining still smaller atoms in the grain of 
sand, and there is evolved, so we say, with growing certainty, the presentation 
of the point as a definite position in space which is one and is incapable of 
further division. But this view is untenable ; we have, it is true, some idea 
how the grain of sand gets smaller and smaller, but only so long as it remains 
just visible; after that we are completely in the dark, and we cannot see or 
imagine the further diminution. That this procedure comes to an end is 
unthinkable ; that nevertheless there exists a term beyond which it cannot go, 
we must believe or postulate without ever reaching it . . . It is a pure 
act of will, not of the understanding." Max Simon observes similarly {Euclid, 
p. 85) "The notion 'point' belongs to the limit-notions (Grenzbegriffe^, the 
necessary conclusions of continued, and in themselves unlimited, senes of 
presentations." He adds, "The point is the limit of localisation; if this is 
more and more energetically continued, it leads to the limit-notion 'point,' 



158 BOOK I [i. deff, r, 2 

better 'position,' which at the same time involves a change of notion. Content 
of space vanishes, relative position remains. 'Point' then, according to our 
interpretation of Euclid, is the extremest limit of that which we can still think 
of (not observe) as a spatial presentation, and if we go further than that, not 
only does extension cease but even relative place, and in this sense the 'part' 
is nothing." I confess I think that even the meaning which Simon intends to 
convey is better expressed by "it has no part" than by "the part is nothing," 
since to take a "part" of a thing in Euclid's sense of the result of a simple 
division, corresponding to an arithmetical fraction, would not be to change 
the notion from that of the thing divided to an entirely different one. 

Definition 2. 

I 'pafip.!} St nrjiu>% a'wXaT«. 

A line is breadthless length. 

This definition may safely be attributed to the Platonic School, if not to 
Plato himself. Aristotle (Topics vi. 6, 143 b 11) speaks of it as open to 
objection because it "divides the genus by negation," length being necessarily 
either breadthless or possessed of breadth ; it would seem however that the 
objection was only taken in order to score a point against the Platonists, since 
he says (ibid. 143 b 29) that the argument is "of service only against those 
who assert that the genus [sc. length] is one numerically, that ts, those who 
assume ideas," e.g. the idea of length (amo u^*o$) which they regard as a 
genus : for if the genus, being one and self-existent, could be divided into 
two species, one of which asserts what the other denies, it would be self- 
contradictory (WaiU), 

Proclus (pp. 96, 21—97, 3) observes that, whereas the definition of a point 
is merely negative, the line introduces the first "dimension," and so its 
definition is to this extent positive, while it has also a negative element which 
denies to it the other " dimensions " (Suur-rdtrtii). The negation of both 
breadth and depth is involved in the single expression "breadthless" (atrXarti), 
since everything that is without breadth is also destitute of depth, though the 
converse is of course not tnie. 

Alternative definitions. 

The alternative definition alluded to by Proclus, fiiyiOtn iip' ty Suumjov 
" magnitude in one dimension " or, better perhaps, " magnitude extended one 
way " (since Suuttoitk as used with reference to line, surface and solid scarcely 
corresponds to our use of " dimension " when we speak of "one," " two," or 
" three dimensions "), is attributed by an-Nairlzs to " Heromides," who must 
presumably be the same as " Herundes," to whom he attributes a certain 
definition of a point. It appears however in substance in Aristotle, though 
Aristotle does not use the adjective Sunmirw, nor does he apparently use 
8«wrrao-« except of body as having three " dimensions " or " having dimension 
(or extension J a// ways (vavrg)," the "dimensions" being in his view (1) up 
and down, (2) before and behind, and (3) right and left, and " up " being the 
principle or beginning of length, " right " of breadth, and " before " of depth 
(De cat So 11. 2, 284 b 24). A line is, according to Aristotle, a magnitude 
" divisible in one way only " (jtattajrg iuLifttrov), in contrast to a magnitude 
divisible in two ways (&XB ZuxtpiTOv), or a surface, and a magnitude divisible 
"in all or in three ways" (iramg koX rpixfi jtnipcToV), or a body (Metaph, 
1016 b 25 — 27); or it is a magnitude "continuous one way (or in one 
direction)," as compared with magnitudes continuous tovo ways or three ways, 



i. def. 2] NOTES ON DEFINITIONS i, 2 159 

which curiously enough he describes as " breadth " and " depth " respectively 

(jtiytdos Zi to p.iv lift tv uvvi^h firjitm, to 8' hrl Suo tAiitos, to 8' twi rpia (3a8o<;, 
Metaph. 1020 a 11), though he immediately adds that " length " means a line, 
" breadth " a surface, and " depth " a body. 

Proclus gives another alternative definition as "flux of a point " (/5ucr« 
tnjfMuiv), i.e. the path of a point when moved. This idea is also alluded to in 
Aristode (De anima 1. 4, 409 a 4 above quoted) : " they say that a line by its 
motion produces a surface, and a point by its motion a line." "This 
definition," says Proclus (p. 97, 8 — r3), "is a perfect one as showing the 
essence of the line : he who called it the flux of a point seems to define it 
from its genetic cause, and it is not every line that he sets before us, but only 
the immaterial line , for it is this that is produced by the point, which, though 
itself indivisible, is the cause of the existence of things divisible." 

Proclus (p. r 00, 5 — 19) adds the useful remark, which, he says, was 
current in the school of Apollonius, that we have the notion of a line when we 
ask for the length of a road or a wall measured merely as length ; for in that 
case we mean something irrespective of breadth, viz. distance in one 
*' dimension." Further we can obtain sensible perception of a line if we look 
at the division between the light and the dark when a shadow is thrown on 
the earth or the moon ; for clearly the division is without breadth, but has 
length. 

Species of "lines." 

After defining the " line " Euclid only mentions one species of line, the 
straight line, although of course another species appears in the definition of a 
circle later. He doubtless omitted all classification of lines as unnecessary for 
his purpose, whereas, for example, Heron follows up his definition of a line by 
a division of lines into (1) those which are " straight " and {2} those which are 
not, and a further division of the latter into (a) " circular circumferences," 
{&) "spiral-shaped" (iiutottScw) lines and (r) "curved" (xajim-vAai) lines generally, 
and then explains the four terms. Aristotle tells us {Metaph. 986 a 25) that 
the Pythagoreans distinguished straight (iJW) and curved (ku/u t',W), and this 
distinction appears in Plato (cf. Republic x. 602 c) and in Aristotle (cf. " to a 
line belong the attributes straight or curved," Anal. post. 1. 4, 73 b 19; "as in 
mathematics it is useful to know what is meant by the terms straight and 
curved," De anima I. 1, 402 b 19). But from the class of " curved " lines 
Plato and Aristotle separate off the urtpi^ipijs or " circular " as a distinct 
species often similarly contrasted with straight. Aristotle seems to recognise 
broken lines forming an angle as one line : thus "a line, if it be bent (xosafi- 
pin}), but yet continuous, is called one" (Me tap A. 1 01 6 a 2); "the straight line 
is more one than the bent line" (Hid. 1016 a 12). Cf. Heron, Def. 12, "A 
broken line (jeckW/u'vij y pappy) so-called is a line which, when produced, 
does not meet itself." 

When Proclus says that both Plato and Aristotle divided lines into those 
which are "straight," "circular" (wtpujxpife) or "a mixture of the two," adding, 
as regards Plato, that he included in the last of these classes " those which are 
called helicoidal among plane (curves) and (curves) formed about solids, and 
such species of curved lines as arise from sections of solids " (p. 104, 1 — 5), 
he appears to be not quite exact. The reference as regards Plato seems to be 
to Parmenides 145 B: "At that rate it would seem that the one must have 
shape, either straight or round (arpoyyvKov) or some combination of the two"; 
but this scarcely amounts to a formal classification of lines. As regards 



i6o 






BOOK I 



[l. DEF. 2 



Aristotle, Proclus seems to have in mind the passage (De each i. 2, 268 b 17) 
where it is stated that " all motion in space, which we call translation ($apd), is 
(in) a straight line, a circle, or a combination of the two ; for the first two ate 
the only simple (motions)." 

For completeness it is desirable to add the substance of Proclus' account 
of the classification of lines, for which he quotes Geminus as his authority. 

Geminus* first classification of lines. 

This begins {p. in, 1 — 9) with a division of lines into composite [avtSrrtK) 
and incomposite (io-u'ec'rrov). The only illustration given of the composite 
class is the "broken line which forms an angle" (9 KftXavjiivy) ™1 ywiW 
ttoumaa) ; the subdivision of the incomposite class then follows (in the text as 
it stands the word " composite " is clearly an error for " incomposite "). The 
subdivisions of the incomposite class are repeated in a later passage (pp. 176, 
27 — 177, 23) with some additional details. The following diagram reproduces 
the effect of both versions as far as possible (all the illustrations mentioned by 
Proclus being shown in brackets). 

lines 



composite 
(broken line forming an angle) 



incomposite 



forming a figure 

GXytHiTtrroiQvaa.i 
or determinate 

(circle, ellipse, eissoid) 



not forming a figure 

or 

indeterminate 

d6pt&Toi 

and 

extending without limit 

i-r Arttpop iKfia.W&fin'a.t 

(straight line, parabola, hyperbola, conchoid) 



The additional details in the second version, which cannot easily be shown 
in the diagram, are as follows : 

(1) Of the lines which extend without limit, some do not form a figure at 
all (viz. the straight line, the parabola and the hyperbola); but some first 
"come together and form a figure" (i.e. have a loop), "and, for the rest, 
extend without limit " (p. 177, 8). 




As the only other curve, besides the parabola and the hyperbola, which 
has been mentioned as proceeding to infinity is the conchoid (of Nicomedes), 
we can hardly avoid the conclusion of Tannery 1 that the curve which has a 
loop and then proceeds to infinity is a variety of the conchoid itself. As is 

1 Notes. pour thistoire des ligncs et surf aits tourbes dans Fatttiquiii in Bulletin des sriemts 
mathim, ct astronam. 1 ser, vm. (1884), pp. 108—0 (Minmires seientifiaues, ri. p. 13). 



I. def. 2] NOTE ON DEFINITION 2 161 

well known, the ordinary conchoid (which was used both for doubling the 
cube and for trisecting the angle) is obtained in this way. Suppose any 
number of rays passing through a fixed point (the pole) and intersecting a 
fixed straight line ; and suppose that points are taken on the rays, beyond the 
fixed straight line, such that the portions of the rays intercepted between the 
fixed straight line and the point are equal to a constant distance (hiacrnj/ia), 
the locus of the points is a conchoid which has the fixed straight line for 
asymptote. If the "distance" a is measured from the intersection of the ray 
with the given straight line, not in the direction away from the pole, but 
towards the pole, we obtain three other curves according as a is less than, 
equal to, or greater than b, the distance of the pole from the fixed straight line, 
which is an asymptote in each case. The case in which a ■■■■ l> gives a curve 
which forms a loop and then proceeds to infinity in the way Proclus describes. 
Now we know both from Eutocius {Csmm. on Archimedes, ed. Heiberg, in. 
p. 98) and Proclus (p. 272, 3 — 7) that Nicomedes wrote on conchoidr (in 
the plural), and Pappus <iv. p. 244, 18) says that besides the "first" (used as 
above stated) there were "the second, the third and the fourth which are 
useful for other theorems." 

(2) Proclus next observes (p. 177, 9) that, of the lines which extend 
without limit, some are "asymptotic" (an^ummK^ namely "those which 
never meet, however they are produced," and some are " symptotic" namely 
" those which will meet sometime " ; and, of the " asymptotic " class, some 
are in one plane, and others not. Lastly, of the "asymptotic" lines in one 
plane, some preserve always the same distance from one another, while others 
continually " lessen the distance, like the hyperbola with reference to the 
straight line, and the conchoid with reference to the straight line." 

Geminus' second classification. 

This (from Proclus, pp. in, 9 — 20 and 112, 16 — 18) can be shown in a 
diagram thus : 

Incomposite lines 
itiimBntx. ypafiftat 

' ' ' ? 

sirnple, drX$ mixed, part 



making a figure indeterminate 

tfXyW *oiou<ra A6p4^TCt 

(e.g. circle) (straight line} 



lines in planes lines on solids 

at ir Toft rrtpttiit 

r; 1 ' 1 

line meeting itself extending without limit 

4 i* afr-tf tfvfi-Ktnwaix rr i-r Airtipar JKpaWofUnf 
(e.g. cissoid) 



lines formed by uctiom tines round solids 

al Mri rii rtyjdi al rtjA ri mpti. 

(e.g. conic sections, spirit curves) (e.g. kdix about 1 sphere or about a cone) 

1 ; L - — 1 

tumotomtrii Hot homfftomertc 

{cylindrical helix) (all othera) 

Notes on classes of "lines" and on particular curves. 

We will now add the most interesting notes found in Proclus with 
reference to the above classifications or the particular curves mentioned. 



16a BOOK I [i. dec. J 

i. Homoeomeric lines. 

By this term (ofUHOfLtpcU) are meant lines which are alike in all parts, so 
that in any one such curve any part can be made to coincide with any other 
part. Proclus observes that these lines are only three in number, two being 
"simple" and in a plane {the straight line and the circle), and the third 
" mixed," {subsisting) " about a solid," namely the cylindrical helix. The 
latter curve was also called the coehlias or eochlion, and its homoeomeric 
property was proved by Apollonius in his work n-cpt tov ko^Xwu (Proclus, 
p. i«5i 5). The fact that there are only three homoeomeric lines was proved 
by Geminus, "who proved, as a preliminary proposition, that, if from a point 
(ami rav trtjuuuv, but on p. 251, 4 &$' ' v ° s <7ij/iei'ou) two straight lines be drawn 
to a homoeomeric line making equal angles with it, the straight lines are 
equal" {pp. 112, 1—113, 3i cf - P- *5'> 3 ~ l 9)- 

2. Mixed lines. 

It might be supposed, says Proclus (p. 105, n), that the cylindrical helix, 
being homoeomerie, like the straight line and the circle, must like them be 
simple. He replies that it is not simple, but mixed, because it is generated by 
two unlike motions. Two like motions, said Geminus, e.g. two motions at the 
same speed in the directions of two adjoining sides of a square, produce a 
simple line, namely a straight line (the diagonal) ; and again, if a straight line 
moves with its extremities upon the two sides of a right angle respectively, 
this same motion gives a simple curve (a circle) for the locus of the middle 
point of the straight line, and a mixed curve (an ellipse) for the locus of any 
Other point on it (p. 106, 3—15). 

Geminus also explained that the term " mixed," as applied to curves, and 
as applied to surfaces, respectively, is used in different senses. As applied to 
curves, "mixing" neither means simple "putting together" (<rw0«7«) nor 
" blending" (upturn). Thus the helix (or spiral) is a " mixed " line, but (1) it 
is not " mixed " in the sense of " putting together," as it would be if, say, part 
of it were straight and part circular, and (2) it is not mixed in the sense of 
" blending," because, if it is cut in any way, it does not present the appearance 
of any simple lines {of which it might be supposed to be compounded, as it 
were). The " mixing " in the case of lines is rather that in which the con- 
stituents are destroyed so far as their own character is concerned, and are 
replaced, as it were, by a chemical combination (feme h avrjj trwi^apfiira to 
ixpa, koI <rvyic€)(y)i.cra.). On the other hand " mixed " surfaces are mixed in 
the sense of a sort of " blending " [xatci «™ Kpaa-iy). For take a cone gene- 
rated by a straight line passing through a fixed point and passing always 
through the circumference of a circle : if you cut this by a plane parallel to 
that of the circle, you obtain a circular section, and if you cut it by a plane 
through the vertex, you obtain a triangle, the " mixed " surface of the cone 
being thus cut into simple lines (pp. 117, 12 — 118, 23). 

3. Spiric curves. 

These curves, classed with conies as being sections of solids, were dis- 
covered by Perseus, according to an epigram of Perseus' own quoted by 
Proclus (p. ii2, t), which says that Perseus found "three lines upon {or, 
perhaps, in addition to) five sections " (rpew ypn/xtta? hA vltrt Totals). 
Proclus throws some light on these in the following passages : 

"Of the spiric sections, one is interlaced, resembling the horse-fetter 
(finroy irflh)) ; another is widened out in the middle and contracts on each 



I. def. a] NOTE ON DEFINITION 2 163 

side (of the middle), a third is elongated and is narrower in the middle, 
broadening out on each side of it" (p. 11 a, 4 — 8). 

" This is the case with the spiric surface ; for it is conceived as generated 
by the revolution of a circle remaining at right angles [to a plane] and turning 
about a point which is not its centre [in other words, generated by the revo- 
lution of a circle about a straight line in its plane not passing through the 
centre]. Hence the spirt takes three forms, for the centre [of rotation] is 
either on the circumference, or within it, or without it. And if the centre of 
rotation is on the circumference, we have the continuous spire (vwixn^t if 
within, the interlaced {iii.wnrk(yu.evi)), and if without, the open (S«x>js). And 
the spiric sections are three according to these three differences" (p. 119, 
8-r 7 ). 

" When the hippopede, which is one of the spiric curves, forms an angle 
with itself, this angle also is contained by mixed lines" (p. 1.27, 1 — 3). 

" Perseus showed for spirics what was their property {a-vinrr^iia) " 

(P- 356. ")■ 

Thus the spiric surface was what we call a tore, or (when open) an anchor- 
ring. Heron (Def. 97) says it was called alternatively spire {uirflpa} or ring 
(xpuur;); he calls the variety in which "the circle cuts itself," not "interlaced," 
but " cross ing-ltse If" (iTrakkaTTUwra). 

Tannery 1 has discussed these passages, as also did Schiaparelli*. It is clear 
that Prochis' remark that the difference in the three curves which he mentions 
corresponds to the difference between the three surfaces is a slip, due perhaps 
to too hurried transcribing from Geminus ; all three arise from plane sections 
of the open anchor-ring. If r is the radius of the revolving circle, a the 
distance of its centre from the axis of rotation, d the distance of the plane 
section (supposed to be parallel to the axis) from the axis, the three curves 
described in the first extract correspond to the following cases : 

(1) d=a~r. In this case the curve is the hippopede, of which the 
lemniscate of Bernoulli is a particular case, namely that in which a = zr. 

The name hippopede was doubtless adopted for this one of Perseus' curves 
on the ground of its resemblance to the hippopede of Eudoxus, which seems to 
have been the curve of intersection of a sphere with a cylinder touching it 
internally. 

(2) a + r>d>a. Here the curve is an oval, 

(3) a > rf> a - r. The curve is now narrowest in the middle. 
Tannery explains the " three lines upon (in addition to) five sections " 

thus. He points out that with the open tore there are two other sections 
corresponding to 

(4) d= a : transition from (2) to (3). 

(5) a- r>d> o, in which case the section consists of two symmetrical 
ovals. 

He then shows that the sections of the closed or continuous tore, corre- 
sponding to a = r, give curves corresponding to (2), (3) and (4) only. Instead 
of (t) and (5) we have only a section consisting of two equal circles touching 
one another. 

On the other hand, the third spire (the interlaced variety) gives three new 
forms, which make a group of three in addition to the first group oifive sections. 

1 Poiw fkisteire des lignts et surfaces conrbts dans tantiquiti in Bulletin tits stientes 
mat him. et astranom. vill. (i88+|, pp. 35— % 7 [Mlnairts stientifiqiw, II. pp. 14— 18). 

s Du kemocentrischen SpAarcn des EudaxuSi ties Kallippus und dts Aristottles {Adhartd* 
lungm tur Gesch. der Math. 1. Heft, 187;, pp. 14Q — <S*)> 



i6 4 



BOOK J 






[l. DF-F. a 



The difficulty which I see in this interpretation is the fact that, just after 
" three lines on five sections " are mentioned, Proclus describes three curves 
which were evidently the most important ; but these three belong to three of 
the five sections of the open tore, and are not separate from them, 

4. The cissoid. 

This curve is assumed to be the same as that by means of which, according 
to Eutocius (Comm. on Archimedes, in. p. 66 sqq.), Diodes in his book ir«pi 
■Kvpiwv (On burning-glasses) solved the problem of doubling the cube. It is 
the locus of points which he found by the following construction. Let AC, 
BD be diameters at right angles in a circle with centre O. 

Let E, Fbe points on the quadrants BC, BA respectively such that the 
arcs BE, BE ait equal. 

Draw EG, FH perpendicular to CA. D 

Join AE, and let P be its intersection 
with FH. 

The cissoid is the locus of all the 
points P corresponding to different posi- 
tions of E on the quadrant BC and of F 
at an equal distance from B along the arc 
BA. 

A ts the point on the curve correspond- 
ing to the position C for the point E, and 
B the point on the curve corresponding 
to the position of E in which it coincides 
with B. 

It is easy to see that the curve extends 
in the direction AB beyond B, and that 
CK drawn perpendicular to CA is an 
asymptote. It may be regarded also as 
having a branch AD symmetrical with 
AB, and) beyond D, approaching KC produced as asymptote. 

If OA, 0£> are coordinate axes, the equation of the curve is obviously 

/(« + *) = ("-*)*> 
where a is the radius of the circle. 

There is a cusp at A, and it agrees with this that Proclus should say 
(p. 1 a 6, 34^ that "cissoidal lines converging to one point like the leaves of 
ivy — for this is the origin of their name — form an angle." He makes the 
slight correction (p. 1 28, 5) that it is not two farts of a curve, but one curve, 
which in this case makes an angle. 

But what is surprising is that Proclus seems to have no idea of the curve 
passing outside the circle and having an asymptote, for he several times 
speaks of it as a closed curve (forming a figure and including an area) : cf. 
p. 152, 7, "the plane (area) cut off by the cissoidal line has one bounding 
(line), but it has not in it a centre such that all (straight lines drawn to the 
curve) from it are equal." It would appear as if Proclus regarded the cissoid 
as formed by the/our symmetrical cissoidal arcs shown in the figure. 

Even more peculiar is Proclus' view of the 

5, "Single-turn Spiral." 

This is really the spiral of Archimedes traced by a point starting from 
the fixed extremity of a straight line and moving uniformly along it, while 





/' 






.--'' ° 


H '•'-.J 






5 




E 


7 


s~"^ 




/ 




j 







!. Dvrr. 2—4] NOTES ON DEFINITIONS 2—4 165 

simultaneously the straight line itself moves uniformly in a plane about its fixed 
extremity. In Archimedes the spiral has of course any number of turns, the 
straight line making the same number of complete revolutions. Yet I'roclus, 
while giving the same account of the generation of the spiral (p. 180, 8 — 12), 
regards the single-turn spiral us actually stopping short at the point reached 
after one complete revolution of the straight line : " it is necessary to know 
that extending without limit is not a property of all lines ; for it neither 
belongs to the circle nor to the cissoid, nor in general to lines which form 
figures ; nor even to those which do not form figures. For even the single- 
turn spiral does not extend without limit— -for it is constructed between two 
points^nor does any of the other lines so generated do so" (p. 187, 19 — 25). 
It is curious that Pappus (vm. p. n 10 sqq. } uses the same term finvoVrpoi^ns 
\\ii to denote one turn, not of the spiral, but of the cylindrical helix. 

Definition 3. 

Tp*Hprj$ SI Tripara {TTj^Mld. 

The extremities of a line art points. 

It being unscientific, as Aristotle said, to define a point as the " extremity 
of a line " (iripas ypa/iftjjs), thereby explaining the prior by the posterior, 
Euclid defined a point differently ; then, as it was necessary to connect a 
point with a line,, he introduced this explanation after the definitions of both 
had been given. This compromise is no doubt his own idea; the same 
thing occurs with reference to a surface and a line as its extremity in Def. 6, 
and with reference to a solid and a surface as its extremity in xt. Def. 2. 

We miss a statement of the facts, equally requiring to be known, that a 
" division " (Suupio-ij) of a line, no less than its " beginning " or " end," is a 
point (this is brought out by Aristotle: cf, Metapk. 1060 b 15), and that 
the intersection of two lines is also a point. If these additional explanations 
had been given, Proclus would have been spared the difficulty which he finds 
in the fact that some of the tines used in Euclid (namelv infinite straight lines 
on the one hand, and circles on the other) have no " extremities." So also 
the ellipse, which Proclus calls by the, old name (foptfc (" shield "). In the 
case of the circle and ellipse we can, he observes (p. 105, 7), take a portion 
bounded by points, and the definition applies to that portion. His rather 
far-fetched distinction between two aspects of a circle or ellipse as a line and 
as a closed figure (thus, while you are describing a circle, you have two extremi- 
ties at any moment, but they disappear when it is finished) is an unnecessarily 
elaborate attempt to establish the literal universality of the "definition," 
which is really no more than an explanation that, if a line has extremities, 
those extremities are points. 

Definition 4. 

Eu#eiu yfnififurj tarty, $rtf ci laov toIs i<f tavriji <r<i(fi(ot* xcirai. 
A straight line is a line which lies evenly with the points on itself. 

The only definition of a straight line authenticated as pre-Euclidean is 
that of Plato, who defined it as " that of which the middle covers the ends " 
(relatively, that is, to an eye placed at either end and looking along the 
straight line). It appears in the Parmenidcs 137 k: "suaight is whatever has 
its middle in front of (i.e. so placed as to obstruct the view of) both its ends " 



1 66 BOOK I [i. dkf. 4 

(diflii yt ov iv to /iitror ap.tj>oir ratv iaxdrotv btiTrpottOfv p). Aristotle quotes it in 
equivalent terms {Topics vi. n, 148 b 27), o5 to /h'o-oc &r«rnotr#«I row inpao-ii' ; 
and, as he does not mention the name of its author, but states it in combina- 
tion with the definition of a line as the extremity of a surface, we may assume 
that he used it as being well known. Proclus also quotes the definition as 
Plato's in almost identical terms, fc to fU<ra row oupow is-tirpoo-fltt (p. 109, ji). 
This definition is ingenious, but implicitly appeals to the sense of sight and 
involves the postulate that the line of sight is straight. (Cf. the Aristotelian 
Problems 31, to, 950 a 39, where the question is why we can better observe 
straightness in a row, say, of letters with one eye than with two.) As regards 
the straightness of "visual rays," oijrui, cf. Euclid's own Optics, Ueff. i, *, 
assumed as hypotheses, in which he first speaks of the " straight lines " drawn 
from the eye, avoiding the word ttyws, and then says that the figure contained 
by the visual rays (oif/us) is a cone with its vertex in the eye. 

As Aristotle mentions no definition of a straight line resembling Euclid's, 
but gives only Plato's definition and the other explaining it as the " extremity 
of a surface," the latter being evidently the current definition in contemporary 
textbooks, we may safely infer that Euclid's definition was a new departure of 
his own. 

Proclus on Euclid's definition. 

Coming now to the interpretation of Euclid's definition, tW«a ypa/1^17 
t<mv, ljTis i£ utou ToTi i$' (ttvnji otjiiiIok KctTat, we find any number of slightly 
different versions, but none that can be described as quite satisfactory ; some 
authorities, e.g. Savile, have confessed that they could make nothing of it It 
is natural to appeal to Proclus first ; and we find that he does in fact give an 
interpretation which at first sight seems plausible. He says {p. 109, 8 sq.) that 
Euclid "shows by means of this that the straight line alone [of all lines] 
occupies a distance (mr^iiv $cacn;/ia) equal to that between the points on it. 
For, as far as one of the points is distant from another, so great is the length 
(p.fytSm) of the straight line of which they are the extremities ; and this is the 
meaning of lying i£ «™v to (or with) the points on it " \i( urov being thus, 
apparently, interpreted as "at" (or "over") "an equal distance"]. "But if 
you take two points on the circumference (of a circle) or any other line, the 
distance cut off between them along the line is greater than the interval 
separating them. And this is the case with every line except the straight line. 
Hence the ordinary remark, based on a common notion, that those who 
journey in a straight line only travel the necessary distance, while those who 
do not go straight travel more than the necessary distance." (Cf. Aristotle, 
De caclo 1. 4, 271 a r3, "we always call the distance of anything the straight 
line" drawn to it.) Thus Proclus would interpret somewhat in this way: "a 
straight line is that which represents extension equal with (the distances 
separating) the points on it." This explanation seems to be an attempt to 
graft on to Euclid's definition the assumption (it is a Xunfiavoptvor, not a 
definition) of Archimedes {On the sphere and cylinder 1. ad init.) that "of all 
the lines which have the same extremities the straight line is least." For this 
purpose i£ «row has apparently to be taken as meaning "at an equal distance," 
and again "lying at an equal distance" as equivalent to "extending over (or 
representing) an equal distance." This is difficult enough in itself, but is 
seen to be an impossible interpretation when applied to the similar definition 
of a plane by Euclid (Def. 7) as a surface "which lies evenly with the straight 
lines on itself." In that connexion Proclus tries to make the same words «£ Iirou 



i. def, 4] NOTE ON DEFINITION 4 167 

■ciroi mean "extends over an equal area with." He says namely (p. 117, 2) 
that, "if two straight lines are set out " on the plane, the plane surface 
"occupies a space equal to that between the straight lines." But two straight 
lines do not determine by themselves any space at all j it would be necessary 
to have a dosed figure with its boundaries in the plane before we could arrive 
at the equivalent of the other assumption of Archimedes that "of surfaces 
which have the same extremities, if those extremities are in a plane, the plane is 
the least [in area]." This seems to be an impossible sense for i£ urov even on 
the assumption that it means " at an equal distance " in the present definition. 
The necessity therefore of interpreting i£ Urov similarly in hoth definitions 
makes it impossible to regard it as referring to distance or length at all. It 
should be added that Simplicius gave the same explanations as Proclus 
(an-Nairtzl, p. 5). 

The language and construction of the definition. 

Let us now consider the actual wording and grammar of the phrase ^ns i£ 
Urov toU i<j> tauTtji <rrj(uioK xttTai. As regards the expression if urov we note 
that Plato and Aristotle (whose use of it seems typical) commonly have it in 
the sense of "on a footing of equality": cf. oi i( EVou in Plato's Laws 777 d, 
gig d; Aristotle, Politics 1259 b g «£ ta-uv ttvat (fovkerai tt/v $va-w t "tend to 
be on an equality in nature," Eth. Nit. vm. 12, 1161 a 8 tVraWa ira>r« i( 
Urov, " there all are on a footing of equality." Slightly different are the uses 
in Aristotle, Eth, Nie. x. 8, 1178 a 25 rur /ikr yap iwyimri'w xP tul *<" i£ Urov 
Itrrta, "both need the necessaries of life to the same extent, Set us say"; Topics ix, 
15, 174 a 32 i$ urov iroioWa rije tpiinjatv, "asking the question indifferently" 
{i.e. without showing any expectation of one answer being given rather than 
another). The natural meaning would therefore appear to be "evenly placed" 
(or balanced), "in equal measure," " indifferently" or "without bias" one way 
or the other. Next, is the dative rots i<f>' io.vrr}t o"i)/u£ok constructed with i( urov 
or with MttBt? In the first case the phrase must mean "that which he&evenfy 
with (or in respect to) the points on it," in the second apparently "that which, 
in (or by) the points on it, lies (or is placed) evenly (or uniformly)." Max Simon 
takes the first construction to give the sense "die Gerade liegt in gleicher 
Weise wie ihre Punkte." If the last words mean " in the same way as (or in 
like manner as) its points," I cannot see that they tell us anything, although 
Simon attaches to the words the notion of distance (Abstand) like Proclus. 
The second construction he takes as giving " die Gerade liegt fur (durch) ihre 
Punkte gleichmassig," " the straight line lies symmetrically for (or through) its 
points"; or, if k«t<h is taken as the passive of Ti'Siff«, "die Gerade ist durch 
ihre Funkte gleichmassig gegeben worden," " the straight line is symmetrically 
determined by its points." He adds that the idea is here direction, and that 
both direction and distance (as between two different given points simply) 
would be to Euclid, as later to Bolzano {Betrachtungen iiber einige Gegenstdnde 
der Elementargeometrie, 1804, quoted by Schotten, Inhalt and Metkode des 
planimetrischtn Unttrrichts, ». p. 16), primary irreducible notions. 

While the language is thus seen to be hopelessly obscure, we can safely 
say that the sort of idea which Euclid wished to express was that of a line 
which presents the same shape at and relatively to all points on it, without 
any irregular or unsymmetrical feature distinguishing one part or side of it 
from another. Any such irregularity could, as Saccheri points out (Engel and 
Stackel, Die Theorie der Parallellinien von Euklidbis Gauss, 1895, p. 109), be 
at once made perceptible by keeping the ends fixed and turning the line about 



1 68 BOOK I [i. def. 4 

them right round; if any two positions were distinguishable, e.g. one being to 
the left or right relatively to another, " it would not lie in a uniform manner 
between its points." 

A conjecture as to its origin and meaning. 

The question arises, what- was the origin of Euclid's definition, or, how 
was it suggested to him ? It seems to me that the basis of it was really 
Plato's definition of a straight line as " that line the middle of which covers 
the ends." Euclid was a Platonist, and what more natural than that he 
should have adopted Plato's definition in substance, while regarding it as 
essential to change the form of words in order to make it independent of any 
implied appeal to vision, which, as a physical fact, could not properly find a 
place in a purely geometrical definition ? I believe therefore that Euclid's 
definition is simply an attempt (albeit unsuccessful, from the nature of the 
case) to express, in terms to which a geometer could not object as not being 
part of geometrical subject-matter, the same thing as the Platonic definition. 

The truth is that Euclid was attempting the impossible. As Pfieiderer 
says (Scholia to Euclid), "It seems as though the notion of a straight line, 
owing to its simplicity , cannot be explained by any regular definition which 
does not introduce words already containing in themselves, by implication, 
the notion to be defined (such e.g. are direction, equality, uniformity or 
evenness of position, unswerving course), and as though it were impossible, if 
a person does not already know what the term straight here means, to teach 
it to him unless by putting before him in some way a picture or a drawing of 
it." This is accordingly done in such books as Veronese's Elementi ii 
geometria (Part I., roo4, p. ro): "A stretched string, e.g. a plummet, a ray of 
light entering by a small hole into a dark room, are rectilineal objects. Hie 
image of them gives us the abstract idea of the limited line which is called a 
rectilineal segment," 

Other definitions. 

We will conclude this note with some other famous definitions of a straight 
line. The following are given by Proclus (p. no, 18 — a 3). 

i, A line stretched to the utmost, bf anpov Ttrafdrr) ypafi./^. This appears 
in Heron also, with the words " towards the ends " (iwl ri urifiara) added. 
(Heron, Def. 4). 

2. Part of it cannot be in the assumed plane while part is in one higher up 
(in fitTiiupoTtpui). This is a. proposition in Euclid (xi. 1). 

3. All its parts fit on all {other parts) alike, Tavra avnjs ™ fii/nj wairiv 
biioiun ty<W*o(f ■. Heron has this too (Def. 4), but instead of " alike " he 
says ravroitos, "in all ways," which is better as indicating that the applied part 
may be applied one way or the reverse way, with the same result. 

4. That line which, when its ends remain fixed, itself remains fixed, ij t«c 
irtpdruiv ittvovrmv naX avr? /itvowra. Heron's addition to this, " when it is, as 
it were, turned round in the same plane " (otur iv ry aw& /n-iW&o orpi^o/tcnf), 
and his next variation, " and about the same ends having always the same 
position," show that thf; definition of a straight line as "that which does 
not change its position when it is turned about its extremities (or any two 
points in it) as poles " was no original discovery of Leibniz, or Saccheri, or 
Krafft, or Gauss, but goes back at least to the beginning of the Christian era. 
Gauss' form of this definition was : " The line in which lie all points that, 
during the revolution of a body (a part of space) about two fixed paints, 
maintain their position unchanged is called a straight line." Schotten 



i. deff. 4, 5J NOTES ON DEFINITIONS 4, 5 169 

(I- p. 315) maintains that the notion of a straight tine and its property of 
being determined by two points are unconsciously assumed in this definition, 
which is therefore a logical "circle." 

5. That line which with one other of tkt same species cannot complete a 
figure, y fA€TCi njs &po<£$ov? /ita; crj^jxa fty dirorfkowsa. This is an obvious 
tcrripov-icponpov, since it assumes the notion of a figure. 

Lastly Leibniz' definition should be mentioned: A straight line is one 
which divides a plane into two halves identical in all but position. Apart from 
the fact that this definition introduces the plane, it does not seem to have any 
advantages over the definition last but one referred to. 

Legendre uses the Archimedean property of a straight line as the shortest 
distance between two points. Van Swinden observes (Elcmcnte der Geometric, 
1834, p. 4), that to take this as the definition involves assuml/tgthe proposition 
that any two sides of a triangle are greater than the third and proving that 
straight lines which have two points in common coincide tnroughout their 
length (cf. Legendre, Aliments de Giom/trie I. 3, 8). 

The above definitions all illustrate the observation of Unger {Die Geometric 
dcs Euilid, 1833) : "Straight is a simple notion, and hence all definitions of 
tt must fail.... But if the proper idea of a straight line has once been grasped, 
it will be recognised in all the various definitions usually given of it ; all 
the definitions must therefore be regarded as explanations, and among them 
that one is the best from which further inferences can immediately be drawn 
as to the essence of the straight line." 

Definition 5. 

Em<ftivtta Si itrroi, $ ft$KM *<" wAaros jaocav Jvtt. 

A surface it that which has length and breadth only. 

The word Iru^drua was used by Euclid and later writers to denote surfact 
in general, while they appropriated the word twiirtSov for plane surface, thus 
making tiriwtSor a species of the genus i-rt^tdyua. A solitary use of hrt$avtta 
by Euclid when a plane is meant (xi. Def. 1 1 ) is probably due to the fact that 
the particular definition came from an earlier textbook. Froclus (p. 116, 17) 
remarks that the older philosophers, including Plato and Aristotle, used the 
words brufrdvtia and imn&ov indifferently for any kind of surface. Aristotle 
does indeed use both words for a surface, with perhaps a tendency to use 
iVi^oKtio more than iV«r«W for a surface not plane. Cf. Categories 6, 5 a 1 sq., 
where both words are used in one sentence : " You can find a common 
boundary at which the parts fit together, a point in the case of a line, and a line 
in the case of a surface (ort^aVtm); for the parts of the surface (imni&ov) do fit 
together at some common boundary. Similarly also in the ease of a body you 
can find a common boundary, a line or a surface (irujuLvftn), at which the 
parts of the body fit together." Plato however does not use ivupdvtia at all in 
the sense of surface, but only cjmrtSov for both surface and plane surface. 
There is reason therefore for doubting the correctness of the notice in 
Diogenes Laertius, 111. a 4, that Plato "was the first philosopher to name, 
among extremities, the//o«# surface" («V»VtSo$ rnufmytui). 

(Tt^ocuo of course means literally the feature of a body which is apparent 
to the eye (iwi^avrj*), namely the surface. 

Aristotle tells us (De sensu 3, 439 a 31) that the Pythagoreans called a 
surface xpoin, which seems to have meant skin as well as colour. Aristotle 
explains the term with reference to colour (xp&l*") a* a thing inseparable from 
the extremity (srepai) of a body. 



i7» BOOK 1 [i. def. s 

Alternative definitions. 

The definitions of a surface correspond to those of a line. As in Aristotle 
a line is a magnitude " (extended) one way, or in one ' dimension ' " (i<j> h), 
"continuous one way" (i<j> tr trunx^), or "divisible in one way" (^iowiy^ 
Statpfroi-), so a surface is a magnitude extended or continuous two ways (hi 
Sw>), or divisible in two ways (Sixi)). As in Euclid a surface has " length and 
breadth " only, so in Aristotle " breadth " is characteristic of the surface and is 
once used as synonymous with it (Metaph. tojo a ix), and again "lengths 
are made up of long and short, surfaces of broad and narrow, and solids (oyi™) 
of deep and shallow" (Metaph. 1085 a 10). 

Aristotle mentions the common remark that a line by its motion produces a 
surface (De anima ]. 4, 409 a 4). He also gives the a posteriori description of 
a surface as the "extremity of a solid" (Topics vi. 4, [41 b 21}, and as "the 
section (rofnf) or division {S«up«rii) of a body" (Metaph. 1060 b 14). 

Proclus remarks (p. 114, jo) that we get a notion of a surface when we 
measure areas and mark their boundaries in the sense of length and breadth ; 
and we further get a sort of perception of it by looking at shadows, since 
these have no depth {for they do not penetrate the earth) but only have length 
and breadth. 

Classification of surfaces. 

Heron gives (Def. 74, p. 50, ed. Heiberg) two alternative divisions or 
surfaces into two classes, corresponding to Gemirtus' alternative divisions of 
lines, viz. into (1) incomposite and composite and (2) simple and mixed. 

(1) Incomposite surfaces are "those which, when produced, fall into (or 
coalesce with) themselves" (wtim iK0akXo/ityai, airat tiaff iavrwv Wwrowie), 
i.e. are of continuous curvature, e.g. the sphere. 

Composite surfaces are "those which, when produced, cut one another." 
Of composite surfaces, again, some are (a) made up of non-homogeneous 
(elements) (i$ dvofiotoyow) such as cones, cylinders and hemispheres, others 
(&) made up of homogeneous (elements), namely the rectilineal (or polyhedral) 
surfaces. 

(2) Under the alternative division, simple surfaces are the plane and the 
spherical surfaces, but no others ; the mixed class includes all other surfaces 
whatever and is therefore infinite in variety. 

Heron specially mentions as belonging to the mixed class (a) the surface 
of cones, cylinders and the like, which are a mixture of plane and circular 
(fitKTal i£ tTrnrt'Sou Kal irepic^epttas) and (h) spirie surfaces, which are "a mixture 
of two circumferences " (by which he must mean a mixture of two circular 
elements, namely the generating circle and its circular motion about an axis in 
the same plane). 

Proclus adds the remark that, curiously enough, mixed surfaces may arise 
by the revolution either of simple curves, e.g. in the case of the spire, or of 
mixed curves, e.g. the "right-angled conoid" from a parabola, "another 
conoid" from the hyperbola, the "oblong" (krifinKts, in Archimedes iropa- 
pa-xts) and " flat " (VwtjrAa-nJ) spheroids from an ellipse according as it revolves 
about the major or minor axis respectively (pp. 119, 6 — 120, 2). The homoeo- 
meric surfaces, namely those any part of which will coincide with any other 
part, are two only (the plane and the spherical surface), not three as in the case 
of lines (p. no, 7). 



i. deff. 6, 7] NOTES ON DEFINITIONS 5— 7 i?» 

Definition 6. 

'Eirt^avtias Si -ripara. ypappmi. 

The extremities of a surface are lines. 

It being unscientific, as Aristotle says, to define a line as the extremity of 
a surface, Euclid avoids the error of defining the prior by means of the 
posterior in this way, and gives a different definition not open to this 
objection. Then, by way of compromise, and in order to show the connexion 
between a line and a surface, he adds the equivalent of the definition of a line 
previously current as an explanation. 

As in the corresponding Def. 3 above, he omits to add what is made 
clear by Aristotle (Metapk. 1060 b 15) that a "division" (Siaipco-is) or 
" section " (to/ii;) of a solid or body is also a surface, or that the common 
boundary at which two parts of a solid fit together (Categories 6, 5 a a) 
may be a surface. 

Proclus discusses how the fact stated in Def. 6 can be said to be true of 
surfaces like that of the sphere "which is bounded (wtiripairrat), it is true, but 
not by lines." His explanation (p. 116, 8 — 14) is that, "if we take the surface 
(of a sphere), so far as it is extended two ways (BtyjJ &a<rr<iTi/), we shall find 
that it is bounded by lines as to length and breadth ; and if we consider the 
spherical surface as possessing a form of its own and invested with a fresh 
quality, we must regard it as having fitted end on to beginning and made 
the two ends (or extremities) one, being thus one potentially only, and not in 
actuality." 

Definition 7, 

'Eir(jr«S« brttJMvtta itrrtv, iJtii t£ itrmi rati 1$ iavrijs ivBtiais wttrai. 
A plane surface is a surface which ties evenly with the straight lines on 
itself. 

The Greek follows exactly the definition of a straight line mutatis mutandis , 
i.e. with Ta«...riS(t'ttit for tok. . .(rryuiW Proclus remarks that, in general, 
all the definitions of a straight line can be adapted to the plane surface by 
merely changing the genus. Thus, for instance, a plane surface is " a surface 
the middle of which covers the ends " (this being the adaptation of Plato's 
definition of a straight line). Whether Plato actually gave this as the defini- 
tion of a plane surface or not, I believe that Euclid's definition of a plane 
surface as lying evenly with the straight lines on itself was intended simply to 
express the same idea without any implied appeal to vision (just as in the 
corresponding case of the definition of a straight line). 

As already noted under Def. 4, Proclus tries to read into Euclid's defini- 
tion the Archimedean assumption that "of surfaces which have the same 
extremities, if those extremities are in a plane, the plane is the least." But, 
as I have stated, his interpretation of the words seems impossible, although it 
is adopted by Simplicius also (see an-Naiiizi). 

Ancient alternatives. 

The other ancient definitions recorded are as follows. 

1. The surf act which is stretched to the utmost («V axpav t tropin}) : a 
definition which Proclus describes as equivalent to Euclid's definition (on 
Proclus' own view of that definition). Cf. Heron, Def. 9, " (a surface) which 
is right (and) stretched out " (ip8i) ovtra airortrnjuoTj), words which he adds to 
Euclid's definition. 



ija BOOK I [l mf. 7 

z. The least surface among all those which have the same extremities. 
Proclus is here (p. 1 1 7, 9) obviously quoting the Archimedean assumption. 

3. A surface all the farts of which have the property of fitting on (each 
ether) (Heron, Def. 9). 

4. A surface such that a straight line fits on all parts of it (Proclus, 
p. 117, 8), or such that the straight line fits on it all ways, i.e. however placed 
(Proclus, p. 117, jo). 

With this should be compared : 

5. "(A plane surface is) such that, if a straight line pass through two 
points on it, the line coincides wholly with it at every spot, all ways," i.e. however 
placed (one way or the reverse, no matter how), 7c ixtiSiiv &vo trqutuiiv aijnjTai 
ritdtla, koX okrj aurp Kara irarra roirctv miPTOtfd? i$apfi4£*Tat (Heron, Def. 9). 

This appears, with the words «<rra mivra Torov iraiToMot omitted, in Theon of 
Smyrna (p. 112, 5, ed. Hiller), so that it goes back at least as far as the 
1st c. a.d. It is of course the same as the definition commonly attributed to 
Robert Simson, and very widely adopted as a substitute for Euclid's. 

This same definition appears also in an-Nairlzl (ed. Curtze, p. 10) who, 
after quoting Simplicius' explanation (on the same lines as Proclus') of the 
meaning of Euclid's definition, goes on to say that "others defined the plane 
surface as that in which it is possible to draw a straight tine from any point to 
any other." 

pifficuitiea in ordinary definitions. 

Gauss observed in a letter to Bessel that the definition of a plane surface 
as a surface such that, if any him points in it be tahcn, the straight line joining 
them lies wAolfy in the surface (which, for short, we will call "Simson's" 
definition) contains more than is necessary, in that a plane can be obtained by 
simply projecting a straight line lying in it from a point outside the line but also 
lying on the plane ; in fact the definition includes a theorem, or postulate, as 
well. The same is true of Euclid's definition of a plane as the surface which 
"lies evenly with (all) the straight lines on itself," because it is sufficient for a 
definition of a plane if the surface " lies evenly " with those lines only which 
pass through a fixed point on it and each of the several points of a straight line 
also lying in it but not passing through the point But from Euclid's point 
of view it is immaterial whether a definition contains more than the necessary 
minimum provided that the existence of a thing possessing all the attributes 
contained in the definition is afterwards proved. This however is not done 
in regard to the plane. No proposition about the nature of a plane as such 
appears before Book XI., although its existence is presupposed in all the 
geometrical Books 1.— iv. and vi. ; nor in Book xi. is there any attempt to 
prove, _e.g. by construction, the existence of a surface conforming to the 
definition. The explanation may be that the existence of the plane as defined 
was deliberately assumed from the beginning like that of points and lines, the 
existence of which, according to Aristotle, must be assumed as principles 
unproved, while the existence of everything else must be proved ; and tt may 
well be that Aristotle would have included plane surfaces with points and 
lines in this statement had it not been that he generally took his illustrations 
from plane geometry (excluding solid). 

But, whatever definition of a plane is taken, the evolution of its essential 
properties is extraordinarily difficult. Crelle, who wrote an elaborate article 
Zur Theorie der Ebene (read in the Academie der Wissenschaften in 1834) of 
which account must be taken in any full history of the subject, observes that, 




I. dep. 7] NOTE ON DEFINITION 7 173 

since the plane is the field, as it were, of almost all the rest of geometry, while 
a proper conception of it is necessary to enable Eucl. 1. 1 to be understood, 
it might have been expected that the theory of the plane would have been the 
subject of at least the same amount of attention as, say, that of parallels. This 
however was far from being the case, perhaps because the subject of parallels 
(which, for the rest, presuppose the notion of a plane) is much taster than that 
of the plane. The nature of the difficulties as regards the plane have also 
been pointed out recently by Mr Frank land {The First Book of Euclid's 
Elements, Cambridge, 1905) ; it would appear that, whatever definition is 
taken, whether the simplest (as containing the minimum necessary to deter- 
mine a plane) or the more complex, e.g. Simson's, some postulate has to be 
assumed in addition before the fundamental properties, or the truth of the 
other definitions, can be established. Crelle notes the same thing as regards 
Simson's definition, containing mare than is necessary. Suppose a plane in 
which lies the triangle ABC. Let AD join the vertex A 
to any point D on EC, and BE the vertex B to any 
point E on CA. Then, according to the definition, AD 
lies wholly in the plane of the triangle; so does BE. 
But, if both AD and BE are to lie wholly in the one 
plane, AD, BE must intersect, say at F: if they did not, 
there would be two planes in question, not one. But the fact that the lines 
intersect and that, say, AD does not pass above or below BE, is by no 
means self-evident 

Mr Frankland points out the similar difficulty as regards the simpler 
definition of a plane as the surface generated by a straight 
line passing always through a fixed point and always 
intersecting a fixed straight line. Let OFF, OQQ 
drawn from intersect the straight line X at P, Q 
respectively. Let E be any third point on X : then it 
needs to be proved that OR intersects P'Q in some 
point, say R'. Without some postulate, however, it is 
not easy to see how to prove this, or even to prove that P'Q intersects X. 

Crelle'a essay. Definitions by Fourier, Deahna, Becker, 
Crelle takes as the standard of a good definition that it shall be, not only as 
simple as possible, but also the best adapted for deducing, with the aid of the 
simplest possible principles, further properties belonging to the thing defined. 
He was much attracted by a very lucid definition, due, he says, to Fourier, 
according to which a flam is formed by the aggregate of ail. the straight lints 
which, passing through one point on a straight line in space, are perpendicular 
to that straight line. (This is really no more than an adaptation from Euclid's 
proposition Xi. 5, to the effect that, if one of four concurrent straight lines be 
at right angles to each of the other three, those three are in one plane, which 
proposition is also used in Aristotle, Meteorohgica m. 3, 373 a 13.) But 
Crelle confesses that he had not been able to deduce the necessary properties 
from this and had had to substitute the definition, already mentioned, of a 
plane as the surface containing, throughout their whole length, all the straight 
lines passing through a fixed point and also intersecting a straight line in space ; 
and he only claims to have proved, after a long series of propositions, that the 
" Fourier "- or " perpendicular "-surface and the plane of the other definition 
just given are identical, after which the properties of the "Fourier "-surface 
can be used along with those of the plane. The advantage of the Fourier 
definition is that it leads easily, by means of the two propositions that 




174 BOOK I |l def. 7 

triangles are equal in all respects (i) when two sides and the included angle 
are respectively equal and (j) when all three sides are respectively equal, to the 
property expressed in Simsou's definition. But Crelle uses to establish these 
two congruence-theorems a number of propositions about equal angles, supple- 
mentary angles, right angles, greater and less angles ; and it is difficult to 
question the soundness of Schotten's criticism that these notions in themselves 
really presuppose that of a plane. The difficulty due to Fourier's use of 
the word " perpendicular," if that were all, could no doubt be got over. Thus 
Deahna in a dissertation (Marburg, 1837) constructed a plane as follows. 
Presupposing the notions of a straight line and a sphere, he observes that, if a 
sphere revolve about a diameter, all the points of its surface which move 
describe closed curves (circles). Each of these circles, during the revolution, 
moves along itself, and one of them divides the surface of the sphere into two 
congruent parts. The aggregate then of the lines joining the centre to the 
points of this circle forms thepfojie. Again, J. K. Becker (Die Elemente der 
Geometric, 1877) pointed out that the revolution of a right angle about one 
side of it produces a conical surface which differs from all other conical 
surfaces generated by the revolution of other angles in the fact that the 
particular cone coincides with the cone vertically opposite to it : this characteristic 
might therefore be taken in order to get rid of the use of the right angle. 

W. Bolyai and Lobachewsky. 

Very similar to Deahna's equivalent for Fourier's definition is the device 
of W. Bolyai and Lobachewsky (described by Frischauf, Elementt der 
absnlulen Geometric, 1876). They worked upon a fundamental idea first 
suggested, apparently, by Leibniz, Briefly stated, their way of evolving a 
plane and a straight line was as follows. Conceive an infinite number of 
pairs of concentric spheres described about two fixed points in space, O, 0, 
as centres, and with equal radii, gradually increasing : these pairs of equal 
spherical surfaces intersect respectively in homogeneous curves (circles), and 
the " InbegrifiT" or aggregate of these curves of intersection forms a plane. 
If A be a point on one of these circles {k say), suppose points M, M' to start 
simultaneously from A and to move in opposite directions at the same speed 
till they meet at B, say ; B then is "opposite" to A, and A, .ff divide the 
circumference into two equal halves. If the points A, B be held fast and the 
whole system be turned about them until O takes the place of 0, and of 
O, the circle k will occupy the same position as before (though turned a 
different way). Two opposite points, P, Q say, of each of the other circles 
will remain stationary during the motion as well as A, B : the " InbegrifT" or 
aggregate of all such points which remain stationary forms a straight line. It 
is next observed that the plane as defined can be generated by the revolution 
of the straight line about 0(7, and this suggests the following construction 
for a plane. Let a circle as one of the curves of intersection of the pairs of 
spherical surfaces be divided as before into two equal halves &tA,B. Let the 
arc ADB be similarly bisected at D, and let C be the 
middle point of AB. This determines a straight line CD 
which is then defined as "perpendicular" to AB. The revo- 
lution of CD about AB generates a plane. The property 
stated in Simson's definition is then proved by means of the 
congruence-theorems proved in Eucl. 1. 8 and 1. 4. The 
first is taken as proved, practically by considerations of 
symmetry and homogeneity. If two spherical surfaces, not necessarily equal, 
with centres O, intersect, A and its "opposite" point B are taken as 





L D:r. 7] NOTE ON DEFINITION 7 175 

before on the curve of intersection (a circle) and, relatively to 00', the point 
A is taken to be convertible with B or any other point on the homogeneous 
curve. The second (that of Eucl. i. 4) is established by simple application. 
Rausenberger objects to these proofs on the grounds that the first assumes 
that the two spherical surfaces intersect in one single curve, not in several, 
and that the second compares angles : a comparison which, be says, is possible 
only in a plane, so that a plane is really presupposed. Perhaps as regards 
the particular comparison of angles Rausenberger is hypercritical ; but it is 
difficult to regard the supposed proof of the theorem of Eucl. 1. 8 as sufficiently 
rigorous (quite apart from the use of the uniform motion of points for the 
purpose of bisecting lines). 

Simson's property is proved from the two congruence-theorems thus. 
Suppose that AB is " perpendicular " (as defined by Bolyai) to two generators 
CM, CN of a plane, or suppose CM, CN respectively to make with AB two 
angles congruent with one another. It is enough to prove that, if P be any 
point on the straight line MN, then CP, just as 
much as CM, CN respectively, makes with AB two 
angles congruent with one another and is therefore 
a generator. We prove successively the congruence 
of the following pairs of triangles : 

ACM, BCM 

ACN, BCN 

AMN, BMN 
AMP, BMP 
ACP, BCP, 

whence the angles A CP, BCP are congruent. 

Other views. 

Enriques and Amaldi (Ekmenti di geometria, Bologna, 1905), Veronese 
(in his Elementi) and Hilbert all assume as a postulate the property stated in 
Simson's definition. But G, Ingrami {Elementi di geometria, Bologna, 1904) 
proves it tn the course of a remarkable series of closely argued proposition 
based upon a much less comprehensive postulate. He evolves the theory of 
the plane from that of a triangle, beginning with a triangle as a mere three-side 
(trilatero), i.e. a frame, as it were. His postulate relates to the three-side and 
is to the effect that each " (rectilineal) segment " joining a vertex to a point of 
the opposite side meets every segment similarly joining each of the other two 
vertices to the points of the sides opposite to them respectively, and, con- 
versely, if a point be taken on a segment joining a vertex to a point of the 
opposite side, and if a straight line be drawn from another vertex to the point 
on the segment so taken, it will if produced meet the opposite side. A 
triangle is then defined as the figure formed by the aggregate of all the 
segments joining the respective vertices of a three-side to points on the 
opposite sides. After a series of propositions, Ingrami evolves a plane as the 
figure formed by the " half straight-lines " which project from an internal point 
of the triangle the points of the perimeter, and then, after two more theorems, 
proves that a plane is determined by any three of its points which are not in 
a straight line, and that a straight line which has two points in a plane has all 
its points in it. 

The argument by which Bolyai and I>obachewsky evolved the plane is 
of course equivalent to the definition of a plane as the locus of all points 
equidistant from two fixed points in space. 



iy6 BOOK I [i. deff, 7—9 

Leibniz in a letter to Giordano defined a plane as thai surface which 
divides space into turn congruent farts. Adverting to Giordano's criticism that 
you could conceive of surfaces and lines which divided space or a plane into 
two congruent parts without being plane or straight respectively, Beez ( liber 
Euklidischc und Nicht-Euklidische Geometric, 1888) pointed out that what was 
wanted to complete the definition was the further condition that the two 
congruent spaces could be slid along each other without the surfaces ceasing 
to coincide, and claimed priority for his completion of the definition in this 
way. But the idea of all the parts of a plane fitting exactly on all other parts 
is ancient, appearing, as we have seen, in Heron, Def. 9. 

Definitions 8, 9. 

8. '£mV«Sos &i yiMivla iariv y iv cwnri&w Svo ypafijjL&v OriTTOfUvtav a\kijktitv 
jcat i*t] tV iiOua.% Ktifiivtiiv TTfm aAAijXuf ruv ypu^/Hav xAurtt. 

9. "Orav &* al Trfptixowai ttjv yutvcW ypaftfXal tv&4itu umtiv, tv&uypaftftos 
jraA.«T<u )J ymria, 

8. A plane angle is the inclination to one another of two lines in a plane 
which meet one another and do not lie in a straight line, 

9. And when the lines containing the angle are straight, the angle is called 
rectilineal. 

The phrase " not in a straight line " is strange, seeing that the definition 
purports to apply to angles formed by curves as well as straight lines. We 
should rather have expected continuous (tTwtxfc) with one another; and 
Heron takes this to be the meaning, since he at once adds an explanation as 
to what is meant by lines not being continuous (oi <rw<x<»)- It looks as though 
Euclid really intended to define a rectilineal angle, but on second thoughts, 
as a concession to the then common recognition of curvilineal angles, altered 
" straight lines " into " lines " and separated the definition into two. 

I think all our evidence suggests that Euclid's definition of an angle as 
inclination (*A£«s) was a new departure. The word does not occur in 
Aristotle ; and we should gather from him that the idea generally associated 
with an angle in his time was rather deflection or breaking of lines (kKiIit^) : cf. 
his common use of xtK\a<r6at and other parts of the verb nASf, and also his 
reference to one bent line forming an angle (tt)i/ ntKapjiivipr vol lyaotrav ftriav 1 
Metaph. 1016 a 13) 

Proclus has a long and elaborate note on this definition, much of which 
(pp. i2i t 12 — 126, 6) is apparently taken direct from a work by his master 
Syrian us (6 iJfiiTtpo! m&vjnpimrL Two criticisms contained in the note need 
occasion no difficulty. One of these asks how, if an angle be an inclination, 
one inclination can produce two angles. The other (p. 1 28, 2) is to the effect 
that the definition seems to exclude an angle formed by one and the same 
curve with itself, e.g. the complete cissoid [at what we call the " cusp "1 or the 
curve known as the hippopede (horse-fetter) [shaped like a lemniscate]. But 
such an "angle" as this belongs to higher geometry, which Euclid may well 
be excused for leaving out of account in any case. 

Other ancient definitions: Apollonius, Plutarch, Carpus. 

Proclus' note records other definitions of great interest. Apollonius 
defined an angle as a contracting of a surface or a solid at one point under a 
broken line Or surface ((rvmyuyq ift^BWMM 4 <7T«p«o5 vpot Iri oij/uiy viti 
<t((cAa<r^tv); ypawji y iruf>arflf), where again an angle is supposed to be 
formed by one broken line or surface. Still more interesting, perhaps, is the 
definition by " those who say that the first distance under the point (ri •wp&rar 



i. Dbff. 8, 9] NOTES ON DEFINITIONS 7—9 177 

&uiim)iia thro to mffuiw) is the angle. Among these is Plutarch, who insists 
that Apollonius meant the same thing ; for, he says, there must be some first 
distance under the breaking (or deflection) of the including lines or surfaces, 
though, the distance under the point being continuous, it is impossible to 
obtain the actual first, since every distance is divisible without limit" (**•' 
ajrttpov). There is some vagueness in the use of the word " distance" (Suwrnj/ia) ; 
thus it was objected that " if we anyhow separate off the first " (distance being 
apparently the word understood) " and draw a straight line through it, we get 
a triangle and not one angle." In spite of the objection, I cannot but see in 
the idea of Plutarch and the others the germ of a valuable conception in 
infinitesimals, an attempt (though partial and imperfect) to get at the rate 
of divergence between the lines at their point of meeting as a measure of the 
angle between them. 

A third view of an angle was that of Carpus of Antioch, who said " that 
the angle was a quantity (too-ov), namely a distance (SuMmf/m) between the 
lines or surfaces containing it This means that it would be a distance (or 
divergence) in one sense (i<fi tv Siurrwc), although the angle is not on that 
account a straight line. For it is not everything extended in one sense (to 1$ tv 
Biao-raToV) that is a line." This very phrase " extended one way" being held 
to define a line, it is natural that Carpus' idea should have been described as 
the greatest possible paradox (wdrrw TrupoSo^orarov). The difficulty seems to 
have been caused by the want of a different technical term to express a new 
idea ; for Carpus seems undoubtedly to have been anticipating the more 
modern idea of an angle as representing divergence rather than distance, and to 
have meant by <+' %v in one sense (rotationatly) as distinct from one way or in 
one dimension (linearly). 

To what category does an angle belong ? 

There was much debate among philosophers as to the particular category 
(according to the Aristotelian scheme) in which an angle should be placed ; 
is it, namely, a quantum (voaov), quale (iroioV) or relation (irpos ti)? 

I. Those who put it in the category of quantity argued from the fact that 
a plane angle is divided by a line and a solid angle by a surface. Since, then, 
it is a surface which is divided by a line, and a solid which is divided by 
a surface, they felt obliged to conclude that an angle is a surface or a solid, and 
therefore a magnitude. But homogeneous finite magnitudes, e.g. plane 
angles, must bear a ratio to one another, or one must be capable of being 
multiplied until it exceeds the other. This is, however, not the case with a 
rectilineal angle and the horn-like angle (jiton™*^), by which tatter is meant 
the "angle" between a circle and a tangent to it, since (Eucl. in. 16) the 
latter "angle" is less than any rectilineal angle whatever. The objection, it 
will be observed, assumes that the two sorts of angl<£ are homogeneous. 
Plutarch and Carpus are classed among those who, in one way or other, placed 
an angle among magnitudes; and, as above noted, Plutarch claimed Apollonius 
as a supporter of his view, although the word contraction (of a surface or solid) 
used by the latter does not in itself suggest magnitude much more than Euclid's 
inclination. It was this last consideration which doubtless led " Aganis," the 
" friend " (socius) apparently of Simplicius, to substitute for Apollonius' 
wording " a quantify which has dimensions and the extremities of which arrive 
at one point" (an-NairUE, p. 13). 

3. Eudemus the Peripatetic, who wrote a whole work on the angle, main- 
tained that it belonged to the category of quality. Aristotle had given as his 
fourth variety of quality "figure and the shape subsisting in each thing, and, 



178 BOOK I [1. Deff. 8, 9 

besides these, straight ness, curvature, and the like " (Categories 8, 10 a 11). 
He says that each individual thing is spoken of as quale in respect of its form, 
and he instances a triangle and a square, using them again later on (ibid. 1 1 a 5) 
to show that it is not all qualities which are susceptible of more and less ; again, 
in Physics 1. 5, 188 a 25 angle, straight, circular are called kinds of figure. 
Aristotle would no doubt have regarded deflection (xtukairQai) as belonging to 
the same category with straightness and curvature (KOf«rvA<mft). At all events, 
Eudemus took up an angle as having its origin in the breaking or deflection 
(«Aao-«) of lines : deflection, he argued, was quality if straightness was, and that 
which has its origin in quality is itself quality. Objectors to this view argued 
thus. If an angle be a quality (n-oionjs) like heat or cold, how can it be bisected, 
say? It can in fact be divided; and, if things of which divisibility is an 
essential attribute are varieties of quantum and not qualities, an angle cannot 
be a quality. Further, the more and the less are the appropriate attributes of 
quality, not the equal and the unequal ; if therefore an angle were a quality, 
we should have to say of angles, not that one is greater and another smaller, 
but that one is more an angle and another less an angle, and that two angles 
are not unequal but dissimilar (ouo'/ioum). As a matter of fact, we are told by 
Simplicius, 538, at, on Arist De caelo that those who brought the angle under 
the category of quale did call equal angles similar angles ; and Aristotle 
himself speaks of similar angles in this sense in De caelo 296020, 311 b 34. 

3. Euclid and all who called an angle an inclination are held by Sy nanus 
to have classed it as a relation (irpoi ti). Yet Euclid certainty regarded angles 
as magnitudes ; this is clear both from the earliest propositions dealing 
specifically with angles, e.g. 1. 9, 13, and also (though in another way) from 
his describing an angle in the very next definition and always as contained 
(wtpiixo/Uvr]') by the two lines forming it (Simon, Euclid, p. 28}. 

Proclus (i.e. in this case Sy nanus) adds that the truth lies between these 
three views. The angle partakes in fact of all those categories : it needs the 
quantity involved in magnitude, thereby becoming susceptible of equality, 
inequality and .the like ; it needs the qualify given it by its form, and lastly 
the relation subsisting between the lines or planes bounding it. 

Ancient classification of " angles." 

An elaborate classification of angles given by Proclus (pp. 126, 7 — 127, 16) 
may safely be attributed to Ge minus. In order to show it by a diagram it 

Angles 

on surfaces in solidt 
{I* OTcptoil) 

, I ' ~ — "i 

on simple surfaces on mixed surfaces 

(e.g. cones, cylinders) 

on planes on spherical surfaces 

1 ' 1 ' 1 

made by simple lines made by "mixed" lines by one of each 

e. g. the angle made by a (e.g. the angle fanned by an 
curve, such as the cissoid ellipse and its axis of by 

and hippppede, with itself) an ellipse and a circle) 

— ' — " 1 T"* '. ] 

ine line.circumf. cLrcumf.-circumf. 



line- eon vex line -concave convex-convex concave-concave mixed, or 

(e.g. angle of a e.g. korH-likt (dp^fcupw) {a^lraAwl convex-concave 

semicircle) {Kcflaroeidfa) or "scraper-lilce" (e.g. those of 

(fwrpotiftii) tunes) 



I. Dkff, 8, 9] NOTE ON DEFINITIONS 8, 9 179 

will be necessary to make a convention about terms. Angles are to be under- 
stood under each class, " line-circumference " means an angle contained by a 
straight line and an arc of a circle, " line-convex " an angle contained by a 
straight line and a circular arc with convexity outwards, and so on in every 
case. 

Definitions of angle classified. 

As for the point, straight line, and plane, so foi the angle, Schotten gives 
a valuable summary, classification and criticism of the different modern views 
up to date (Inhalt und Methode des planitnetrisehen Unterrichts, 11., 1893, 
pp. 94 — 183} ; and for later developments represented by Veronese reference 
may be made to the third article (by Amaldi) in Questioni riguardanti le 
matematiche elementari, t, (Bologna, 19 12). 

With one or two exceptions, says Schotten, the definitions of an angle may 
be classed in three groups representing generally the following views : 

1. The angle is the difference of direction between two straight lines. (With 
this group may be compared Euclid's definition of an angle as an inclination.) 

2. The angle is the quantity or amount (or the measure) of the rotation 
necessary to bring one of its sides from its own position to thai of the other s:de 
without its moving out of the plane containing both. 

3. The angle is the portion of a plane included between two straight tines in 
the plane which meet in a point (or two rays issuing from the point). 

It is remarkable however that nearly all of the text-books which give 
definitions different from those in group 2 add to them something pointing to 
a connexion between an angle and rotation : a striking indication that the 
essential nature of an angle is closely connected with rotation, and that a good 
definition must take account of that connexion. 

The definitions in the first group must be admitted to be (autologous, or 
circular, inasmuch as they really presuppose some conception of an angle. 
Direction (as between tow given points) may no doubt be regarded as a primary 
notion ; and it may be defined as "the immediate relation of two points which 
the ray enables us to realise" (Schotten). But "a direction is no intensive 
magnitude, and therefore two directions cannot have any quantitative 
difference " (Biirklen). Nor is direction susceptible of differences such as 
those between qualities, e.g. colours. Direction is a singular entity : there 
cannot be different sorts or degrees of direction. If we speak of "a different 
direction," we use the word equivocally ; what we mean is simply " another " 
direction. The fact is that these definitions of an angle as a difference of 
direction unconsciously appeal to something outside the notion of direction 
altogether, to some conception equivalent to that of the angle itself. 

Recent Italian views. 

The second group of definitions are (says Amaldi) based on the idea of the 
rotation of a straight line or ray in a plane about a point : an idea which, 
logically formulated, may lead to a convenient method of introducing the 
angle. But it must be made independent of metric conceptions, or of the 
conception of congruence, so as to bring out first the notion of an angle, and 
afterwards the notion of equal angles. 

The third group of definitions satisfy the condition of not including metric 
conceptions ; but they do not entirely correspond to our intuitive conception 
of an angle, to which we attribute the character of an entity in one dimension 
(as Veronese says) with respect to the ray as element, or an entity in two 



i8o BOOK I [l Dec 9 

dimensions with reference to feints as elements, which may be called an angular 
sector. The defect is however easily remedied by considering the angle as 
" the aggregate of the rays issuing from the vertex and comprised in the angular 
sector," 

Proceeding to consider the principal methods of arriving at the logical 
formulation of the first superficial properties of the plant from which a 
definition of the angle may emerge, Amaldi distinguishes two points of view 
(l) the genetic, (z) the actual. 

(i) From the first point of view we consider the cluster of straight lines 
or rays (the aggregate of all the straight lines in a plane passing through a 
point, or of all the rays with their extremities in that point) as generated by 
the movement of a straight line or ray in the plane, about a point. This leads 
to the post ulation of a closed order, or circular disposition, of the straight lines 
or rays in a cluster. Next comes the connexion subsisting between the 
disposition of any two clusters whatever in one, plane, and so on. 

(2) Starting from the point of view of the actual, we lay the foundation 
of the definition of an angle in the division of the plane into two parts (half- 
planes) by the straight line. Next, two straight lines (a, b) in the plane, inter- 
secting at a point O, divide the plane into four regions which ate called 
angular sectors (convex) ; and finally the angle (ab) or (6a) may lie defined as 
the aggregate of the rays issuing from O and belonging to the angular sector 
which has a and b for sides. 

Veronese's procedure (in his Elementi) is as follows. He begins with the 
first properties of the plane introduced by the following definition. 

The figure given by all the straight lines joining the points of a straight 
line r to a point P outside it and by 
the parallel to r through P is called a 
cluster of straight lines, a cluster of rays, 
or a plane, according as we consider 
the element of the figure itself to be the 
straight line, the ray terminated at P, 
or a foinl. 

[It will be observed that this method of producing a plane involves using 
the parallel to r. This presents no difficulty to Veronese because he has 
previously defined parallels, without reference to the plane, by means of reflex 
or opposite figures, with respect to a point O : " two straight lines are called 
parallel, if one of them contains two points opposite to (or the reflex of) two 
points of the other with respect to the middle point of a common transversal 
(of the two lines)." He proves by means of a postulate that the parallel r 
does belong to the plane Pr, Ingrami avoids the use of the parallel by 
defining a plane as "the figure formed by the half straight lines which project 
from an internal point of a triangle (i.e. a point on a line joining any vertex of 
a three-side to a point of the opposite side) the points of its perimeter," and 
then defining a cluster of rays as "the aggregate of the half straight lines in a 
plane starting from a given point of the plane and passing through the points 
of the perimeter of a triangle containing the point"] 

Veronese goes on to the definition of an angle. " We call an angle a part 
of a cluster of rays, bounded by two rays (as the segment is a part of a straight 
line bounded hy two points). 

"An angle of the cluster, the bounding rays, of which are opposite, is called a 
flat angle." 

Then, after a postulate corresponding to postulates which he lays down for 




i. Deff. 9-1 a] NOTES ON DEFINITIONS 9—12 181 

a rectilineal segment and for a straight line, Veronese proves that ail flat angles 
are equal to one another. 

a v e 




Hence he concludes that "the cluster of rays is a homogeneous linear 
system in which the element is the ray instead of the point. The cluster 
being a homogeneous linear system, all the propositions deduced from 
[Veronese's] Post. 1 for the straight line apply to it, e.g. that relative to 
the sum and difference of the segments : it is only necessary to substitute 
the ray for the point, and the angle for the segment." 

Definitions 10, ti, 12. 

id. T Orak Si cvdfui hr iv&tlav trra$tura tu? tfj>e£y<> yoivta? urac cIAA^'Xacs 
TOtp, Ap&i) itcaripa i£r law ywetuie fart, Hal ij l$t<rri}m/ta tv#«a itdOtrm KoXiirai, 

Ijt TjV ilflilTTTJKfV, 

1 1 . A/i/JAeiJi yavia itrriv )J ft«'£o»' ipftjs. 

1 2 . '0£tia Si 1} tXA&irmr ipftjs. 

10. When a straight line set up on a straight line makes the adjacent angles 
equal is one another, each of the equal angles is right, and the straight line 
standing on the other is called a perpendicular to that on which it stands. 

11. An obtuse angle is an angle greater than a right angle. 
1 2. An acute angle is an angle less than a right angle. 

i4>t^<s is the regular term for adjacent angles, meaning literally " (next) in 
order." I do not find the term used in Aristotle of angles, but he explains its 
meaning in such passages as Physics VI. 1, 131 b 8 : "those things are (next) 
in order which have nothing of the same kind (tjvyy&h) between them." 

KaSrrtn, perpendicular, means literally let fall: the full expression is perpen- 
dicular straight line, as we see from the enunciation of Eucl. 1. 11, and the 
notion is that of a straight line let fall upon the surface of the earth, s. plumb- 
line. Proclus (p. 283, 9) tells us that in ancient times the perpendicular was 
called gnomon-wise (Kara yecJporo), because the gnomon (an upright stick) was 
set up at right angles to the horizon. 

The three kinds of angles are among the things which according to the 
Platonic Socrates {Republic vi. 510 c) the geometer assumes and argues from, 
declining to give any account of them because they are obvious. Aristotle 
discusses the priority of the right angle in comparison with the acute (Metaph. 
1084 b ;): in one way the right angle is prior, i.e. in being defined (on 
(Jptirrat) and by its notion (t<^ Aoyijt), in another way the acute is prior, i.e. as 
being a part, and because the right angle is divided into acute angles ; the 
acute angle is prior as matter, the right angle in respect of form ; cf. also 
Metaph. 1035 b 6, "the notion of the right angle is not divided into 



i8a BOOK I [i. Dt.tr. 12-14 

that of an acute angle, but the reverse ; for, when denning an acute angle, 
you make use of the right angle." Proclus {p. 133, 15) observes that it is by 
the perpendicular that we measure the heights of figures, and that it is by 
reference to the right angle that we distinguish the other rectilineal angles, 
which are otherwise undistinguished the one from the other. 

The Aristotelian Problems {16, 4, 013 b 36) contain an expression perhaps 
worth quoting. The question discussed is why things which fall on the 
ground and rebound make "similar" angles with the surface on both sides of 
the point of impact ; and it is observed that " the right angle is the limit 
(opm) of the opposite angles," where however " opposite " seems to mean, not 
" supplementary " (or acute and obtuse), but the equal angles made with the 
surface on opposite sides of the perpendicular. 

Proclus, after his manner, remarks that the statement that an angle less 
than a right angle is acute is not true without qualification, for ( 1 ) the horn-like 
angle (between the circumference of a circle and a tangent) is less than a 
right angle, since it is less tnan an acute angle, but is not an acute angle, while 
(2) the "angle of a semicircle" (between the arc and a diameter) is also less 
than a right angle, but is not an acute angle. 

The existence of the right angle is of course proved in 1. n. 

Definition 13. 

*Opos iariv, tivos lati iripas. 

A boundary is that which is an extremity of anything. 

Aristotle also uses the words Spot and W/kk as synonymous. Cf. De gen. 
animal, it. 6, 745 a 6, 9, where in the expression '* limit of magnitude " first 
one and then the other word is used. 

Proclus {p, 136, 8) remarks that the word boundary is appropriate to the 
origin of geometry, which began from the measurement of areas of ground 
and involved the marking of boundaries. 

Definition 14. 

%-^rjyni Itrrt to vtt6 tlvo* ij WW Qpwv Trtptt^ofitvot: 
A figure is that which is contained Ay any boundary or boundaries. 
Plato in the Meno observes that roundness (o-rpoyyvkarrfi) or the round is a 
" figure," and that the straight and many other things are so too j he then 
inquires what there is common to all of them, in virtue of which we apply the 
term "figure" to them. His answer is (76 a): "with reference to every 
figure I say that that in which the solid terminates (tovto, tU S to <rrtp€oy 
mpaivft) is a figure, or, to put it briefly, a figure is an extremity of a solid." 
The first observation is similar to Aristotle's in the Physics 1, 5, 188 a 25, 
where angle, straight, and circular are mentioned as genera of figure. In the 
Categories 8, 10 a 11, "figure" is placed with straightness and curved ness in 
the category of quality. Here however " figure " appears to mean shape 
(ttopifnj) rather than " figure " in our sense. Coming nearer to ''figure' 1 in our 
sense, Aristotle admits that figure is "a sort of magnitude" {fit anima ill. 1, 
425 a i£), and he distinguishes plane figures of two kinds, in language not 
unlike Euclid's, as contained by straight and circular lines respectively ! "every 
plane figure is either rectilineal or formed by circular lines (xtpufitp6ypa.ii.por), 
and the rectilineal figure is contained by several lines, the circular by one 
line" (De caelo 11. 4, 286 b rj). He is careful to explain that a plane is not a 



i. Deff. 14-16] NOTES ON DEFINITIONS it— 16 183 

figure, nor a figure a plane, but that a plane figure constitutes one notion and 
is a spaiti of the genus figure {Anal, pott. 11. 3, 00 b 37). Aristotle does not 
attempt to define figure in general, in fact he says it would be useless : " From 
this it is clear that there is one definition of soul in the same way as there is 
one definition of figure; for in the one case there is no figure except the 
triangle, quadrilateral, and so on, nor is there any soul other than those above 
mentioned. A definition might be constructed which should apply to all 
figures but not specially to any particular figure, and similarly with the 
species of soul referred to. [But such a general definition would serve no 
purpose.] Hence it is absurd here as elsewhere to seek a general definition 
which wjll not be properly a definition of anything in existence and will not 
be applicable to the particular irreducible species before us, to the neglect of 
the definition which is so applicable" (De anima ir. 3, 414 b 20 — 28}. 

Comparing Euclid's definition with the above, we observe that by intro- 
ducing boundary (opm) he at once excludes the straight which Aristotle classed 
as figure ; he doubtless excluded angle also, as we may judge by (1) Heron's 
statement that " neither one nor two straight lines can complete a figure," 
(a) the alternative definition of a straight line as "that which cannot with 
another line of the same species form a figure," (3) Ge minus' distinction 
between the line which forms a figure (o-xij/urrosrotowa) and the line which 
extends indefinitely (tr axttpav J*,9oAA<i/iiifi), which latter term includes a 
hyperbola and a parabola. Instead of calling figure an extremity as 
Plato did in the expression "extremity (or limit) of a solid," Euclid 
describes a figure as that -which has a boundary or boundaries. And lastly, 
in spite of Aristotle's objection, he does attempt a general definition to 
cover all kinds of figure, solid and plane. It appears certain therefore that 
Euclid's definition is entirely his own. 

Another view of a figure, recalling that of Plato in Mens 76 a, is attributed 
by Proclus (p. 143, 8) to Posidonius. The latter regarded the figure as the 
confining extremity or limit {ripat avyiAaor), " separating the notion of figure 
from quantity (or magnitude) and making it the cause of definition, limitation, 
and inclusion (rot lipitrSai not TrtrtpatrBtu not t^i irtpioy^). . . Posidonius thus 
seems to have in view only the boundary placed round from outside, Euclid 
the whole content, so that Euclid will speak of the circle as a figure in 
respect of its whole plane (surface) and of its inclusion (from) without, whereas 
Posidonius (makes it a figure) in respect of its circumference... Posidonius 
wished to explain the notion of figure as itself limit trig and confining magnitude." 

Proclus observes that a logical and refining critic might object to Euclid's 
definition as defining the genus from the species, since that which is enclosed 
by one boundary and that which is enclosed by several are both species of 
figure. The best answer to this seems to be supplied by the passage of 
Aristotle's De anima quoted above. 

Definitions 15, 16. 

15. KtikXof Itrrl irjfijfMi t(b(8w iuro pJas ypa/i/i^s Trtpt(xo/t*fov [ij Kakturat 
rtpi^tip€LtA t irpo9 %v ££' £fot trrjfitlov rvv crros rov oyrnfjuiTos KttfAtvwv iratrat at 
irpc*T7ri777ov&ai tvOtlai (Vpos -ryv tqv kvkXov Tr*pt$ipiiav] urtu dXAirfXaif titriv. 

16. Kivrpov Si rot kvk\ov to trijfAfioy xaAfirat. 

15. A circle is a plane figure contained by one line such that all the straight 
lines falling upon it from one point among those lying -within the figure are equal 
to one another ; 

16. And the point is called the centre of the circle. 



184 BOOK I [1. Deff. 15, 16 

The words ij KaXtirm rtpi^iptia, " which is called the circumference," and 
Tpot ttjv toS kvkXov wtpuftipttay, " to the circumference of the circle," are 
bracketed by Heiberg because, although the mss. have them, they are 
omitted in other ancient sources, viz. Proclus, Taurus, Sextus Empiricus and 
Boethius, and Heron also omits the second gloss. The recently discovered 
papyrus Hercuianensis No. 1061 also quotes the definition without the words 
in question, confirming Heiberg's rejection of them (see Heiberg in Hermes 
xxxviii., 1903, p. 47), The words were doubtless added in view of the 
occurrence of the word "circumference" in Deff. 17, 18 immediately 
following, without any explanation. But no explanation was needed. Though 
the word irtpi^iptia does not occur in Flato, Aristotle uses it several times 
(1) in the general sense of con tour without any special mathematical signification, 
{2) mathematically, with reference to the rainbow and the circumference, as 
well as an arc, of a circle. Hence Euclid was perfectly justified in employing 
the word in Deff. 17, 18 and elsewhere, but leaving it undefined as being a 
word universally understood and not involving in itself any mathematical 
conception. It may be added that an-Nairizi had not the bracketed words 
in his text ; for he comments on and tries to explain Euclid's omission to 
define the circumference. 

The definition itself contained nothing new in substance. Plato {Parme- 
nides 137 e) says : " Round is, I take it, that the extremes of which are every 
way equally distant from the middle " (aTpoyyvkor yi irov Am touto, oi &v to 
lo^ara irayra^ airo rov pitrnv Ixrov lirri^tj). In Aristotle we find the following 
expressions : " the circular (v4pi<i>fp6ypajifu>v) plane figure Dounded by one 
line" {De taeh n. 4, 286 b 13 — 16); "the plane equal (i.e. extending equally 
all ways) from the middle " (hrUtfov to i*. tov p.iao\s urov), meaning a 
circle (Rhetoric 111. 6, 1407 b 27); he also contrasts with the circle "any 
other figure which has not the lines from the middle equal, as for example an 
egg-shaped figure" {De eaelo n. 4, 287 a 19). The word "centre" {nivrpov) 
was also regularly used : cf. Produs' quotation from the " oracles " (Wyto), 
" the centre from which all (lines extending) as far as the rim are equal." 

The definition as it stands has no genetic character. It says nothing as to 
the existence or non-existence of the thing defined or as to the method of 
constructing it. It simply explains what is meant by the word " circle," and 
is a provisional definition which cannot be used until the existence of circles 
is proved or assumed. Generally, in such a case, existence is proved by 
actual construction ; but here the possibility of constructing the circle as 
defined, and consequently its existence, are postulated (Postulate 3). A genetic 
definition might state that a circle is the figure described when a straight line, 
always remaining in one plane, moves about one extremity as a fixed point 
until it returns to its first position (so Heron, Def. ay). 

Simplicius indeed, who points out that the distance between the feet of a 
pair of compasses is a straight line from the centre to the circumference, will 
have it that Euclid intended by this definition to show how to construct a 
circle by the revolution of a straight line about one end as centre ; and an- 
Nairlzi points to this as the explanation (r) of Euclid's definition of a circle 
as a plane figure, meaning the whole surface bounded by the circumference, 
and not the circumference itself, and (2) of his omission to mention the 
" circumference," since with this construction the circumference is not drawn 
separately as a line. But it is not necessary to suppose that Euclid himself 
did more than follow the traditional view ; for the same conception of the 
circle as a plane figure appears, as we have seen, in Aristotle. While, however, 



i. Deff. 15-17] NOTES ON DEFINITIONS 15—17 185 

Euclid is generally careful to say the "circumference of a circle " when he means 
the circumference, or an arc, only, there are cases where "circle" means 
"circumference of a circle," e.g. in ill. 10 1 "A circle does not cut a circle 
in more points than two." 

Heron, Proclus and Simplicius are all careful to point out that the centre 
is not the only point which is equidistant from all points of the circumference. 
The centre is the only point in the plane of the circle ("lying within the figure," 
as Euclid says) of which this is true; any point not in the same plane which 
is equidistant from all points of the circumference is a pole. If you set up a 
"gnomon " (an upright stick) at the centre of a circle (i.e. a line through the 
centre perpendicular to the plane of the circle), its upper extremity is a pole 
(Proclus, p. 153, 3); the perpendicular is the locus of all such poles. 

Definition 17. 

ktAfitrpof Si rov kvkKuv ttrrlv tvOttOr tls &ia tw Ktvrpov Tjyfj&vrj k<u irtpaiav- 
p-ivT) ($ tKfXTfpa tq pip7j uiro tt|s ro? kvkXov ircpM^rpfiav, Tfrc? *a>. Stya Ti/ivfi tov 
kvx\qv. 

A diameter of the circle is any straight line drawn through the centre and 
terminated in both directions by the circumference of the circle, and such a straight 
line also bisects the circle. 

The last words, literally " which (straight line) also bisects the circle," 
are omitted by Simson and the editors who followed him. But they are 
necessary even though they do not " belong to the definition " but only 
express a property of the diameter as defined. For, without this explanation, 
Euclid would not have been justified in describing as a .fcrjw-circle a portion 
of a circle bounded by a diameter and the circumference cut off by it. 

Simplicius observes that the diameter is so called because it passes through 
the whole surface of a circle as if measuring it, and also because it divides the 
circle into two equal parts. He might however have added that, in general, it 
is a line passing through a figure where it is widest, as well as dividing it 
equally : thus in Aristotle ri nara Sia/ttrpw KtCptva, " things diametrically 
situated " in space, are at their maximum distance apart. Diameter was the 
regular word in Euclid and elsewhere for the diameter of a square, and also 
of a parallelogram; diagonal (Stayuptot) was a later term, defined by Heron 
(Def. by] as the straight line drawn from an angle to an angle. 

Proclus (p. 157, 10) says that Thales was the first to prove that a circle is 
bisected by its diameter; but we are not told how he proved it. Proclus gives 
as the reason of the property "the undeviating course of the straight line 
through the centre " (a simple appeal to symmetry), but adds that, if it is 
desired to prove it mathematically, it is only necessary to imagine the diameter 
drawn and one part of the circle applied to the other ; it is then clear that 
they must coincide, for, if they did not, and one fell inside or outside the 
other, the straight lines from the centre to the circumference would not all be 
equal : which is absurd. 

Saccheri's proof is worth quoting. It depends on three " Lemmas " 
immediately preceding, (1) that two straight lines cannot enclose a space, 
(z) that two straight lines cannot have one and the same segment common, 
(3) that, if two straight lines meet at a point, they do not touch, but cut one 
another, at it. 

" Let MDHNKM be a circle, A its centre, MN a diameter. Suppose 




i86 BOOK I [i. Deff. 17, tS 

the portion MNKM of the circle turned about the fixed points M, N, so 
that it ultimately comes near to or coincides with the remaining portion 
MNHDM. 

"Then (i) the whole diameter MAN', with all 
its points, clearly remains in the 3ame position, 
since otherwise two straight lines would enclose a 
space (contrary to the first Lemma). 

" (ii) Clearly no point K of the circumference 
NKM falls within or outside the surface enclosed 
by the diameter MANand the other part, NHDM, 
of the circumference, since otherwise, contrary to 
the nature of the circle, a radius as AK would be 
less or greater than another radius as All. 

" (iii) Any radius MA can clearly be rectilineally produced only along a 
single other radius AN, since otherwise (contrary to the second Lemma) two 
lines assumed straight, e.g. MAN, MAH, would have one and the same 
common segment. 

" (iv) All diameters of the circle obviously cut one another in the centre 
(Lemma 3 preceding), and they bisect one another there, by the general 
properties of the circle. 

" From all this it is manifest that the diameter MAN divides its circle 
and the circumference of it just exactly into two equal parts, and the same 
may be generally asserted for every diameter whatsoever of the same circle ; 
which was to be proved." 

Simson observes that the property is easily deduced from tit. 31 and 24 ; 
for it follows from lit. 31 that the two parts of the circle are "similar 
segments" of a circle (segments containing equal angles, in. Def. 11), and 
from tti. 24 that they are equal to one another. 



Definition 18. 

H/iuruKXiov hi itTTt to Trtptt^ofitvov tr)(fffia. inrd re t?/s BiafUTpov Kat Trj% 
Ivokafi^awo/Ltv^^ iiir avr^e vtpi^tpiia^. nivTfujv Si tov ijfiiKvtt\{oii to avrd, & 

Ktll TOV KVtikrtv CffTtV. 

A semicircle if the figure contained by the diameter and the circumference cut 
off by it. And the centre of the semicircle is the same as that of the circle. 

The last words, "And the centre of the semicircle is the same as that 
of the circle," are added from Proclus to the definition as it appears in the 
MSS. Scarburgh remarks that a semicircle has no centre, properly speaking, 
and thinks that the words are not Euclid's, but only a note by Proclus. I am 
however inclined to think that they are genuine, if only because of the very 
futility of an observation added by Proclus. He explains, namely, that the 
semicircle is the only plane figure that has its centre on its perimeter (!), "so 
that you may conclude that the centre has three positions, since it may be 
within the figure, as in the case of a circle, or on the perimeter, as with the 
semicircle, or outside, as with some conic lines (the single-branch hyperbola 
presumably)" ! 

Proclus and Simplicius point out that, in the order adopted by Euclid for 
these definitions of figures, the first figure taken is that bounded by on* line 
(the circle), then follows that bounded by two lines (the semicircle), then the 
triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes 



I. Deff. 18-ai] NOTES ON DEFINITIONS 17—31 187 

different kinds of figures bounded by two lines (pp. 159, 14 — 160, 9). Thus 
they may be formed 

(1) by circumference and circumference, e.g. (a) those forming angles, as 
a tunc (to pijvottSn) and the figure included by two arcs with convexities 
outward, and (b) the angle-less (iytinov), as the figure included between two 
concentric circles (the coronal) ; 

(2) by circumference and straight line, e.g. the semicircle or segments of 
circles (tty£S<t is a name given to those less than a semicircle); 

(3) by " mixed " line and " mixed " line, e.g. two ellipses cutting one 
another ; 

(4) by " mixed " line and circumference, e.g. intersecting ellipse and 
circle; 

(5) by " mixed" line and straight line, e.g. half an ellipse. 

Following Def. 18 in the mss. is a definition of a segment of a circle which 
was obviously interpolated from in. Def. 6. Proclus, Martianus Capella and 
Boethius do not give it in this place, and it is therefore properly omitted. 

Definitions 19, 20, 21. 

19. Sx^fOTo tiSvypa/ifia i<m ra wro tifituSv vepn^optva, rptirXtvpa fiir 
ra viro rpcwv, TtrpawXtvpa Si Ttt vro Tftrtrapwv, Trokvirktvpa Si to Wd ir\<toi*iii' 9 
Ttwdpwv (vfyii'vi- irtpttxoprtva, 

20. Tw Si Tparktvpttiv tTxypATwv Uroirktvpov piy tprMMW cart to rat rptts 
uras tX 01 ' irAtvpas, to-oo-«€ Acs Si to ras Svo povas tcras %\ov irXtvpai, tTKttX.i/fvov Si 
TO rac rptls dvurow; t\oc vktvpos. 

21. *Ett Si Tali' Tpnrktvpwv &)ri]pf£-rwv 6p$oytovtav ptv rpiytovov itrri to l)(ov 
6p$ijv yomav, dp.ftkvytii'tQV Si to tyov dfiflkiiav ytortav, o^vycuviO*' Si to Tat Tpits 
o£cia¥ i%Qv ytin-tas. 

19. Rectilineal figures are those which are contained by straight lines, 
trilateral figures being those contained by three, quadrilateral those contained by 

four, and multilateral those contained by more than four straight lines. 

20. Of trilateral figures, an equilateral triangle is that which has its three 
sides equal, an isosceles triangle that which has two of its sides alone equal, and 
a scalene triangle that which has Us three sides unequal. 

x 1 . Further, of trilateral figures, a right-angled triangle it that which has 
a right angle, an obtuse-angled triangle that which has an obtuse angle, and an 
acute-angled triangle that which has its three angles acute. 

1 9. 

The latter part of this definition, distinguishing three-sided, four-sided and 
many-sided figures, is probably due to Euclid himself, since the words 
TplirXtvpov, TtrpairKivpov and ircAvTrAtvpop do not appear in Plato or Aristotle 
(only in one passage of the Mechanics and of the Problems respectively does 
even rtrpdirktvpov, quadrilateral, occur). By his use of TtTpa'irAtupoK, 
quadrilateral, Euclid seems practically to have put an end to any ambiguity 
in the use by mathematicians of the word Terpdyiiivov, literally "four-angled 
(figure)," and to have got it restricted to the square. Cf. note on Def. 22, 

20. 

Isosceles (io-octmX^, with equal legs) is used by Plato as well as Aristotle. 
ScaJene (o-praAijvos, with the variant o-mtA-irtTje) is used by Aristotle of a triangle 
with no two sides equal : cf. also Tim. Locr. 98 b. Plato, Euthyphro 1 2 rj, 




i88 BOOK I [i. Deff. 20, 21 

applies the term " scalene " to an odd number in contrast to " isosceles " used 
of an even number, l'roclus (p. 168, 24) seems to connect it with tntatu, to 
limp ; others make it akin to a-ioAto's, crooked, aslant. Apollonius uses the 
same word " scalene " of an oblique circular cone. 

Triangles are classified, first with reference to their sides, and then with 
reference to their angles. l'roclus points out that seven distinct species of 
triangles emerge: (1) the equilateral triangle, (2) three species of isosceles 
triangles, the right-angled, the obtuse-angled and the acute-angled, (3} the 
same three varieties of scalene triangles. 

Proclus gives an odd reason for the dual classification according to sides 
and angles, namely that Euclid was mindful of the fact that it is not every 
triangle that is trilateral also. He explains this statement by reference 
(p. 165, 2j) to a figure which some called barb-like (&«SmiSiji) while 
Zenodovus called it hollow-angled ((toiAoywi'w). Proclus mentions it again 
in his note on 1. 22 (p. 328, 21 sqq.) as one of the paradoxes of geometry, 
observing that it is seen in the figure of that proposition. This " triangle " is 
merely a quadrilateral with a re-entrant angle ; and the idea that 
it has only three angles is due to the non-recognition of the 
fourth angle (which is greater than two right angles) as being an 
angle at all. Since Proclus speaks of the four-sided triangle as 
" one of the paradoxes in geometry," it is perhaps not safe to 
assume that the misconception underlying the expression existed 
in the mind of Proclus alone ; but there does not seem to be any evidence 
that Zenodorus called the figure in question a triangle (cf. Pappus, ed. 
Hultsch, pp. 1 154, 1206). 

Definition 22. 

Twv Si TrrpawXcvpwv trj(jjfjidTmv Ttrpdymvov p\kv i<mv, S itroTrXtvpoy t4 lart 
Kai 6p$oy<wtov, irtpofjiTjitts £i, u 6pBoywViov firv, ovx urowktvpov S«, pojiftot S(. S 
UronXtvpov piv, oIk &p$<rywvutr Si, pop.fio*i£is Bi to t&s i.Trtvarrioir vktvpdx rt xal 
yinvia^ icac oAAij'Aats ^X ov > * our* i<jQTr\cvpov i<rriv out* ApGoyvviov To 04 Trapa 
TauTa TfrpdirXtvpa TpaW£ia naKtta&w. 

0/ quadrilateral figures, a square is that which is both equilateral and right- 
angled; an oblong thai which is right-angled but not equilateral; a rhombus 
that which is equilateral but not right-angled ; and a rhomboid that which has 
its opposite sides and angles equal te one another but is neither equilateral nor 
right-angled. And let quadrilaterals ether than these be called trapezia. 

Ttrpayiiivov was already a square with the Pythagoreans (cf, Aristotle, 
Metapk. 986 a 26), and it is so most commonly in Aristotle ; but in Dt anima 
11. 3, 414 t> 31 it seems to be a quadrilateral, and in Metapk. 1054 b 2, 
" equal and equiangular Ttrpdyaiva," it cannot be anything else but quadri- 
lateral if "equiangular" is to have any sense. Though, by introducing 
Tcrpdir\tvpav for any quadrilateral, Euclid enabled ambiguity to be avoided, 
there seem to be traces of the older vague use of Ttrpdytowr in much later 
writers. Thus Heron (Def. 100) speaks of a cube as "contained by six equi- 
lateral and equiangular Ttrpdywra" and Proclus (p. 166, 10) adds to his 
remark about the " four-sided triangle " that " you might have rtrpaybiva with 
more than the four sides," where rtrpayvsva can hardly mean squares. 

tTtpofiyKn, oblong (with sides of different length), is also a Pythagorean term. 

The word right-angled {ipBoyiiytor) as here applied to quadrilaterals 
must mean rectangular (i.e., practically, having all its angles right angles) ; 
for, although it is tempting to take the word in the same sense for a 



I. Def. u] NOTES ON DEFINITIONS so— a j 189 

square as for a triangle (i.e. " having one right angle "), this will not do in the 
case of the oblong, which, unless it were stated that three of its angles are 
right angles, would not be sufficiently defined. 

If it be objected, as it was by Todhunter for example, that the definition 
of a square assumes more than is necessary, since it is sufficient that, being 
equilateral, it should have one right angle, the answer is that, as in other cases, 
the superfluity does not matter from Euclid's point of view ; on the contrary, 
the more of the essential attributes of a thing that could be included in its 
definition the better, provided that the existence of the thing defined and its 
possession of all those attributes is proved before the definition is. actually 
used ; and Euclid does this in the case of the square by construction in 1. 46, 
making no use of the definition before that proposition. 

The word rhombus (po/t/Jot) is apparently derived from fiipfioi, to turn 
round and round, and meant among other things a spinning-top. Archimedes 
uses the term solid rhombus to denote a solid figure made up of two right 
cones with a common circular base and vertices turned in opposite directions. 
We can of course easily imagine this solid generated by spinning; and, if the 
cones were equal, the section through the common axis would be a plane 
rhombus, which would also be the apparent form of the spinning solid to the 
eye. The difficulty in the way of supposing the plane figure to have been 
named after the solid figure is that in Archimedes the cones forming the solid 
are not necessarily equal. It is however possible that the solid to which the 
name was originally given was made up of two equal cones, that the plane 
rhombus then received its name from that solid, and that Archimedes, in 
taking up the old name again, extended its signification (cf. J. H. T. Miiller, 
Beitrdge zur Terminologie der griechisehen Mathematiker, i860, p. 20). 
Proclus, while he speaks of a rhombus as being like a shaken, i.e. deformed, 
square, and of a rhomboid as an oblong that has been moved, tries to explain 
the rhombus by reference to the appearance of a spinning square {rsrpaymvov 
poftfioiittvor). 

It is true that the definition of a rhomboid says more than is necessary in 
describing it as having its opposite sides and angles equal to one another. 
The answer to the objection is the same as the answer to the similar objection 
to the definition of a square. 

Euclid makes no use in the Elements of the oblong, the rhombus and 
the rhomboid. The explanation of his inclusion of definitions of these 
figures is no doubt that they were taken from earlier text- books. From 
the words "let quadrilaterals other than these be called trapezia" we may 
perhaps infer that trapezium was a new name or a new application of an old 
name. 

As Euclid has not yet defined parallel lines and does not anywhere 
define a parallelogram, he is not in a position to make the more elaborate 
classification of quadrilaterals attributed by Proclus to Posidonius and 
appearing also in Heron's Definitions. It may be shown by the following 
diagram, distinguishing seven species of quadrilaterals. 

Quadrilaterals 



parallelograms non- parallelograms 



rectangular non-rectangular two sides parallel no sides parallel 

[traptxtum] {traprt&t) 



r-U 



squart oblong rhombus rhomboid i iGscdts trapezium scaitnt trapezium 



190 BOOK I [1. Bepf. 2s, a3 

It will be observed that, while Euclid in the above definition classes as 
trapezia all quadrilaterals other than squares, oblongs, r ho in hi, and rhomboids, 
the word is in this classification restricted to quadrilaterals having two sides 
(only) parallel, and trapezoid is used to denote the rest Euclid appears to 
have used trapezium in the restricted sense of a quadrilateral with two sides 
parallel in his book, vtpi Suuptatav (on divisions of figures). Archimedes 
uses it in the same sense, but in one place describes it more precisely as a 
trapezium with its two sides parallel. 

Definition 23. 

IlapaX^7]\ai turiv ciStitu, a*riMf iv to) uurw hrtwi&o ov&at mil «*/3aXAo/j.*i'ai 
tit irnpov itji' imiTtpa. Ta /Mpf in ftrfiirtpa tni/iviirTOtxTiv &XXykiuf. 

Parallel straight tines are straight lines which, being in the same plane and 
being produced indefinitely in both directions, do not meet one another in either 
direction. 

Wap6XKvj\\K (alongside one another) written in one word does not appear 
in Plato ; but with Aristotle it was already a familiar term. 

«(? aire tpok cannot be translated " to infinity " because these words might 
seem to suggest a region or place infinitely distant, whereas •« airttpoy, which 
seems to be used indifferently with «jt' Zirnpuv, is adverbial, meaning "without 
limit," i.e. "indefinitely." Thus the expression is used of a magnitude being 
"infinitely divisible," or of a series of terms extending without limit 

in both directions, i<ft' tKattpa ™ fitpift literally "towards both the parts" 
where "parts" must be used in the sense of "regions" (cf Thuc. 11. 96). 

It is clear that with Aristotle the general notion of parallels was that of 
straight lines which do not meet, as in Euclid : thus Aristotle discusses the 
question whether to think that parallels do meet should be called a 
geometrical or an ungeometrical error (Anal. post. 1. 12, 77 b 22), and (more 
interesting still in relation to Euclid) he observes that there is nothing 
surprising in different hypotheses leading to the same error, as one might 
conclude that parallels meet by starting from the assumption, either (a) that 
the interior (angle) is greater than the exterior, or (b) that the angles of a 
triangle make up more than two right angles (Anal, prior, u. 17, 66 a 11). 

Another definition is attributed by Prod us to Posidonius, who said that 
"parallel tines are those which, (being) m one plane, neither converge nor diverge, 
but have all the perpendiculars equal which are drawn from the points oj one 
line to tlit other, while such (straight lines) as make the perpendiculars less and 
less continually do converge to one another ; for the perpendicular is enough 
to define (opt'f %%v jvrarcu) the heights of areas and the distances between lines. 
For this reason, when the perpendiculars are equal, the distances between the 
straight lines are equal, but when they become greater and less, the interval is 
lessened, and the straight lines converge to one another in the direction in 
which the less perpendiculars are " (Proclus, p. 176, 6 — 17). 

Posidonius' definition, with the explanation as to distances between straight 
lines, their convergence and divergence, amounts to the definition quoted by 
Simplicius (an-Nairlzi, p. 25, ed. Curt/.e) which described straight lines as 
parallel if, when they are produced indefinitely both ways, the distance between 
them, or the perpendicular drawn from either of them to the other, is always 
equal and not different. To the objection that it should be proved that the 
distance between two parallel lines is the perpendicular to them Simplicius 



I. Dtr. j 3 ] NOTES ON DEFINITIONS 21, 13 191 

replies that the definition will do equally well if all mention of the perpen- 
dicular be omitted and it be merely stated that the distance remains equal, 
although " for proving the matter in question it is necessary to say that one 
straight line is perpendicular to both" (an-NairizI, ed. Besthorn-Heiberg, p. 9). 
He then quotes the definition of "the philosopher Aganis": "Parallel 
straight tines are straight lints, situated in the same plane, the distance between 
which, if they are predated indefinitely in both directions at the same time, is 
everywhere the same," (This definition forms the basis of the attempt of 
"Aganis" to prove the Postulate of Parallels.) On the definition Simplicius 
remarks that the words "situated in the same plane" are perhaps unnecessary, 
since, if the distance between the lines is everywhere the same, and one does 
not incline at all towards the other, they must for that reason be in the same 
plane. He adds that the "distance" referred to in the definition is the 
shortest tine which joins things disjoined. Thus, between point and point, 
the distance is the straight line joining them ; between a point and a straight 
line or between a point and a plane it is the perpendicular drawn from the point 
to the tine or plane; "as regards the distance between two lines, that distance 
is, if the lines are parallel, one and the same, equal to itself at al! places on 
the lines, it is the shortest distance and, at all places on the lines, perpendicular 
to both" {ibid, p, 10). 

The same idea occurs in a quotation by Proclus (p, 177, n) from 
Geminus. As part of a classification of lines which do not meet he observes : 
" Of lines which do not meet, some are in one plane with one another, others 
not. Of those which meet and are in one plane, some are always the same 
distance from one another, others lessen the distance continually, as the hyper- 
bola (approaches) the straight line, and the conchoid the straight line (i.e. the 
asymptote in each case). For these, while the distance is being continually 
lessened, are continually (in the position of) not meeting, though they converge 
to one another; they never converge entirely, and this is the most paradoxical 
theorem in geometry, since it shows that the convergence of some lines is non- 
convergent. But of lines which are always an equal distance apart, those 
which are straight and never make the (distance) between them smaller, and 
which are in one plane, are parallel." 

Thus the equidistance-theary of parallels (to which we shall return) is very 
fully represented in antiquity. I seem also to see traces in Greek writers of a 
conception equivalent to the vicious direction-theory which has been adopted 
in so many modem text-books. Aristotle has an interesting, though obscure, 
allusion in Anal, prior, ti. 16, 65 a 4 to a petit io principii committed by "those 
who think that they draw parallels " (or " establish the theory of parallels," 
which is a possible translation of t&s «-apaAA»J*ovs ypdifttw) : "for they un- 
consciously assume such things as it is not possible to demonstrate if parallels 
do not exist" It is clear from this that there was a vicious circle in the then 
current theory of parallels ; something which depended for its truth on the 
properties of parallels was assumed in the actual proof of those properties, 
e.g. that the three angles of a triangle make up two right angles. This is not 
the case in Euclid, and the passage makes it clear that it was Euclid himself 
who got rid of the petitio principii in earlier text-books by formulating and 
premising before 1. 19 the famous Postulate 5, which must ever be regarded 
as among the most epoch-making achievements in the domain of geometry. 
But one of the commentators on Aristotle, Philoponus, has a note on the 
above passage purporting to give the specific character of the petitio principii 
alluded to; and it is here that a direction-theory of parallels may be hinted at, 



i9i BOOK I [i. Def. 12 

whether Philoponus is or is not right in supposing that this was what Aristotle 
had in mind. Philoponus says: "The same thing is done by those who draw 
parallels, namely begging the original question ; for they will have it that it is 
possible to draw parallel straight lines from the meridian circle, and they 
assume a point, so to say, falling on the plane of that circle and thus they 
draw the straight lines. And what was sought is thereby assumed; for he 
who does not admit the genesis of the parallels will not admit the point 
referred to either." What is meant is, I think, somewhat as follows. Given 
a straight line and a point through which a parallel to it is to be drawn, we 
are to suppose the given straight line placed in the plane of the meridian. 
Then we are told to draw through the given point another straight line in the 
plane of the meridian (strictly speaking it should be drawn in a plane parallel 
to the plane of the meridian, but the idea is that, compared with the sue of 
the meridian circle, the distance between the point and the straight line is 
negligible) ; and this, as I read Philoponus, is supposed to be equivalent to 
assuming a very distant point in the meridian plane and joining the given 
point to it. But obviously no ruler would stretch to such a point, and the 
objector would say that we cannot really direct a straight line to the assumed 
distant point except by drawing it, without more ado, parallel to the given 
straight line. And herein is the pditio principii. I am confirmed in seeing 
in Philoponus an allusion to a direction-theory by a remark of Schotten on a 
similar reference to the meridian plane supposed to be used by advocates of 
that theory. Schotten is arguing that direction is not in itself a conception 
such that you can predicate one direction of two different lines. " If any one 
should reply that nevertheless many lines can be conceived which all have the 
direction from north to south," he replies that this represents only a nominal, 
not a real, identity of direction. 

Coming now to modern times, we may classify under three groups 
practically all the different definitions that have been given of parallels 
(Schotten, op. cit. it p. 188 sqq.). 

(i) Parallel straight lines have no point common, under which general 
conception the following varieties of statement may be included : 

(a) they do not cut one another, 

(6) they meet at infinity, or 

(c ) they have a common point at infinity. 

(a) Parallel straight lines have the same, or like, direction or directions, 
under which class of definitions must be included all those which introduce 
transversals and say that the parallels make equal angles with a transversal. 

(3) Parallel straight lines have the distance between them constant; 
with which group we may connect the attempt to explain a parallel as the 
geometrical focus of all points which are equidistant from a straight line. 

But the three points of view have a good deal in common ; some of them 
lead easily to the others. Thus the idea of the lines having no point common 
led to the notion of their having a common point at infinity, through the 
influence of modem geometry seeking to embrace different cases under one 
conception ; and then again the idea of the lines having a common point at 
infinity might suggest their having the same direction. The " non-secant " 
idea would also naturally lead to that of equidistance (3}, since our 
observation shows that it is things which come nearer to one another that 
tend to meet, and hence, if lines are not to meet, the obvious thing is to see 
that they shall not conic nearer, i.e. shall remain the same distance apart. 



L Def. i 3 ] NOTE ON DEFINITION 13 193 

We will now take the three groups in order. 

(1) The first observation of Schotten is that the varieties of this group 
which regard parallels as (a) meeting at infinity or (6) having a common 
point at infinity (first mentioned apparendy by Kepler, 1604, as a " facpn de 
parler " and then used by Desargues, 1630) are at least unsuitable definitions 
for elementary text-books. How do we know that the lines cut or meet at 
infinity ? We are not entitled to assume either that they do or that they do 
not, because "infinity" is outside our field of observation and we cannot verify 
either. As Gauss says (letter to Schumacher), " Finite man cannot claim to 
be able to regard the infinite as something to be grasped by means of ordinary 
methods of observation." Steiner, in speaking of the rays passing through a 
point and successive paints of a straight tine, observes that as the point of 
intersection gets further away the ray moves continually in one and the same 
direction (" nach einer und derselben Richtung hin ") ; only in one position, 
that in which it is parallel to the straight line, "there is no real cutting" 
between the ray and the straight line ; what we have to say is that the ray is 
"directed towards the infinitely distant point on the straight line." It is true 
that higher geometry has to assume that the lines do meet at infinity: whether 
such lines exist in nature or not does not matter (just as we deal with "straight 
lines " although there is no such thing as a straight line). But if two lines do 
not cut at any finite distance, may not the same thing be true at infinity also? 
Are lines conceivable which would not cut even at infinity but always remain 
at the same distance from one another even there? Take the case of a line 
of railway. Must the two rails meet at infinity so that a train could not stand 
on them there (whether we could see it or not makes no difference)? It 
seems best therefore to leave to higher geometry the conception of infinitely 
distant points on a line and of two straight lines meeting at infinity, like 
imaginary points of intersection, and, for the purposes of elementary geometry, 
to rely on the plain distinction between " parallel " and " cutting " which 
average human intelligence can readily grasp. This is the method adopted 
by Euclid in his definition, which of course belongs to the group (1) of 
definitions regarding parallels as non-secant. 

It is significant, I think, that such authorities as Ingram! {Elementi di 
geometria, 1904) and Enriques and Amaldi {EUmenti di geometria, 1905), 
after all the discussion of principles that has taken place of late years, give 
definitions of parallels equivalent to Euclid's : " those straight lines in a plane 
which have not any point in common are called parallels." Hilbert adopts 
the same point of view, Veronese, it is true, takes a different line. In his 
great work Eondamenti di geometria, 1891, he had taken a ray to be parallel to 
another when a point at infinity on the second is situated on the first ; but he 
appears to have come to the conclusion that this definition was unsuitable for 
his Element i. He avoids however giving the Euclidean definition of parallels 
as "straight lines in a plane which, though produced indefinitely, never meet," 
because no one has ever seen two straight lines of this sort," and because 
the postulate generally used in connexion with this definition is not evident in 
the way that, in the field of our experience, it is evident that only one straight 
line can pass through two points. Hence he gives a different definition, for 
which he claims the advantage that it is independent of the plane. It is 
based on a definition of figures "opposite to one another with respect to a 
point" (or rtflex figures). "Two figures are opposite to one another with 
respect to a point O, e.g. the figures ABC ... and A'ffC .,., if to every point 
of the one there corresponds one sole point of the other, and if the segments 



194 BOOK I [1. Dbf. 13 

OA, OB, OC, ... joining the points of one figure to O are respectively equal 
and opposite to the segments OA\ OB, OC",... joining to O the corresponding 
points of the second " : then, a transversal of two straight lines being any 
segment having as its extremities one point of one line and one point of the 
other, " two straight lines are called parallel if one of them contains two points 
opposite to two paints of the other with respect to the middle point of a common 
transversal" It is true, as Veronese says, that the parallels so defined and the 
parallels of Euclid are in substance the same ; but it can hardly be said that 
the definition gives as good an idea of the essential nature of parallels as does 
Euclid's. Veronese has to prove, of course, that his parallels have no point in 
common, and his "Postulate of Parallels" can hardly be called more evident 
than Euclid's : " If two straight lines are parallel, they are figures opposite to 
one another with respect to the middle points of all their transversal segments." 

(2) The direction-theory. 

The fallacy of this theory has nowhere been more completely exposed 
than by C. L. Dodgson (Euclid and his modern Rivals, 1879). According to 
Killing (JEinfiihrung in die Grundlagen der Geometrie, 1. p. 5) it would appear 
to have originated with no less a person than Leibniz. In the text-books 
which employ this method the notion of direction appears to be regarded as a 
primary, not a derivative notion, since no definition is given. But we ought 
at least to know how the same direction or like directions can be recognised 
when two different straight lines are in question. But no answer to this 
question is forthcoming. The fact is that the whole idea as applied to non- 
coincident straight lines is derived from knowledge of the properties of 
parallels ; it is a case of explaining a thing by itself. The idea of parallels 
being in the same direction perhaps arose from the conception of an angle as 
a difference of direction (the hoi low n ess of which has already been exposed) ; 
sameness of direction for parallels follows from the same " difference of 
direction " which both exhibit relatively to a third line. But this is not 
enough. As Gauss said ( Werke, iv. p. 365), " If it [identity of direction] is 
recognised by the equality of the angles formed with one third straight line, 
we do not yet know without an antecedent proof whether this same equality 
will also be found in the angles formed with a fourth straight line " {and any 
number of other transversals) ; and in order to make this theory of parallels 
valid, so far from getting rid of axioms such as Euclid's, you would have to 
assume as an axiom what is much less axiomatic, namely that " straight lines 
which make equal corresponding angles with a certain transversal do so with 
any transversal " (Dodgson, p. 101). 

(3) In modern times the conception of parallels as equidistant straight 
lines was practically adopted by Clavius (the editor of Euclid, born at 
Bamberg, 1537) and (according to Saccheri) by Borelli (Euctides restitutus, 
1658) although they do not seem to have defined parallels in this way. 
Saccheri points out that, before such a definition can be used, it has to 
be proved that " the geometrical locus of points equidistant from a straight 
line is a straight line." To do him justice, Clavius saw this and tried to 
prove it : he makes out that the locus is a straight line according to the 
definition of Euclid, because " it lies evenly with respect to all the points 
on it"; but there is a confusion here, because such "evenness" as the locus 
has is with respect to the straight line from which its points are equidistant, 
and there is nothing to show that it possesses this property with respect 
to itself. In fact the theorem cannot be proved without a postulate. 



I. Post, i] NOTE ON POSTULATE t 195 

Postulate i. 

'Hmjff&n aire jravres aijfuiou irl woe (njpMOV tvSiiav ypaptpTJv dyaytiv. 
Let the following be postulated : to draw a straight line from any point to 
any point. 

From any point to any point. In general statements of this kind 
the Greeks did not say, as we do, " any point," " any triangle " etc., but 
"every point," " every triangle " and the like. Thus the words are here 
literally "from every point to every point." Similarly the first words of 
Postulate 3 are " with every centre and distance," and the enunciation, e.g., of 
1. 18 is " In every triangle the greater side subtends the greater angle." 

It will be remembered that, according to Aristotle, the geometer must in 
general assume what a thing is, or its definition, but must prove that it is, 
i.e. the existence of the thing corresponding to the definition : only in the case 
of the two most primary things, points and lines, does he assume, without 
proof, both the definition and the existence of the thing defined. Euclid has 
indeed no separate assumption affirming the existence oi points such as we find 
nowadays in text-books like those of Veronese, Ingrami, Enriques, "there exist 
distinct points" or "there exist an infinite number of points." But, as re- 
gards the only lines dealt with in the Elements, straight lines and circles, 
existence is asserted in Postulates 1 and 3 respectively. Postulate 1 however 
does much more than (1) postulate the existence of straight lines. It is 
(2) an answer to a possible objector who should say that you cannot, with the 
imperfect instruments at your disposal, draw a mathematical straight line at all, 
and consequently (in the words of Aristotle, A/ml. post. 1. to, 76 b 41) that 
the geometer uses false hypotheses, since he calls a line a foot long when it is 
not or straight when it is not straight. It would seem (if Gherard's translation 
is right) that an-Nairlsd saw that one purpose of the Postulate was to refute 
this criticism : "the utility of the first three postulates is (to ensure) that the 
weakness of our equipment shall not prevent (scientific) demonstration " 
(ed. CurUe, p. 30). The fact is, as Aristotle says, that the geometer's demon- 
stration is not concerned with the particular imperfect straight line which he 
has drawn, but with the ideal straight line of which it is the imperfect 
representation. Simplidus too indicates that the object of the Postulate is 
rather to enable the drawing of a mathematical straight line to be imagined 
than to assert that it can actually be realised in practice : " he would be a 
rash person who, taking things as they actually are, should postulate the 
drawing of a straight line from Aries to Libra." 

There is still something more that must be inferred from the Postulate 
combined with the definition of a straight line, namely {3) that the straight 
line joining two points is unique : in other words that, if two straight lines 
("rectilineal segments," as Veronese would call them) have the same extremities, 
they must coincide throughout their length. The omission of Euclid to state 
this in so many words, though he assumes it in 1. 4, is no doubt answerable for 
the interpolation in the text of the equivalent assumption that two straight 
lines cannot enclose a space, which has constantly appeared in mss. and editions 
of Euclid, either among Axioms or Postulates. That Postulate 1 included it, 
by conscious implication, is even clear from Proclus' words in his note on 1. 4 
(p. 139, 16) : "therefore two straight lines do not enclose a space, and it was 
with knowledge of this fact that the writer of the Elements said in the first of 
his Postulates, to draw a straight line from any point to any point, implying 
that it is one straight line which would always join the two points, not two." 




t9 6 BOOK I [i. Post, i, * 

Proclus attempts in the same note (p. 339) to prove that two straight lines 
cannot enclose a space, using as his basis the definition of the diameter of a 
circle and the theorem, stated in it, that any diameter divides the circle into 
two equal parts. 

Suppose, he says, ACS, ADB to be two straight lines enclosing a space. 
Produce them (beyond B) indefinitely. With centre £ 
and distance AB describe a circle, cutting the lines so 
produced in F, E respectively. 

Then, since ACBF, ADBE are both diameters 
cutting off semi-circles, the arcs AE, AEF are equal : 
which is impossible. Therefore etc 

It will be observed, however, that the straight lines 
produced are assumed to meet the circle given in two 
different points E, F, whereas, for anything we know, 
E, F might coincide and the straight lines have three common points. The 
proof is therefore delusive. 

Saccheri gives a different proof. From Euclid's definition of a straight 
line as that which lies evenly with its points he infers that, when 
such a line is turned about its two extremities, which remain fixed, 
all the points on it must remain throughout in the same position, and 
cannot take up different positions as the revolution proceeds. "In 
this view of the straight line the truth of the assertion that two 
straight lines do not enclose a space is obviously involved. In fact, 
if two lines are given which enclose a space, and of which the two 
points A and X are the common extremities, it is easily shown that 
neither, or else only one, of the two lines is straight." 

It is however better to assume as a postulate the (act, inseparably 
connected with the idea of a straight line, that there exists only one straight 
line containing two given points, or, if two straight lines have two points in 
common, they coincide throughout. 

Postulate 2. 

Kai vertpao-iiinpr ivOtlav Kara to minuet <V tiStiai infiaXtuf. 

To produce a finite straight line continuously in a straight line. 

I translate ir«r tpaapi vyv by finite, because that is the received equivalent, 
and because any alternative word such as limited, terminated, if applied to a 
straight line, would equally fail to express what modem Italian geometers aptly 
call a rectilineal segment, that is, a straight line having two extremities. 

Just as Post. 1 asserting the possibility of drawing a straight line from any 
one point to another must be held to declare at the same time that the 
straight line so drawn is unique, so Post. 2 maintaining the possibility of 
producing a finite straight line (a " rectilineal segment ") continuously in a 
straight line must also be held to assert that the straight line can only be 
produced in one way at either end, or that the produced part in either 
direction is unique; in other words, that two straight lines cannot have a 
common segment, This latter assumption is not expressly appealed to by 
Euclid until XL t. But it is needed at the very beginning of Book 1. Proclus 
(p. 114, 18) says that Zeno of Sidon, an Epicurean, maintained that the very 
Erst proposition 1. 1 requires it to be admitted that " two straight lines cannot 
have the same segments " ; otherwise AC, BC might meet before they arrive 
at C and have the rest of their length common, in which case the actual 
triangle formed by them and A B would not be equilateral. The assumption 
that two straight lines cannot have a common segment is certainly necessary 
in 1. 4, where one side of one triangle is placed on that side of the other 



i. Post, a] 



NOTES ON POSTULATES i, 2 



197 




triangle which is equal to it, and it is inferred that the two coincide throughout 

their length : this would by no means follow if two straight lines could have a 

common segment. Proclus (p. 315, 24), while observing that Post. 2 clearly 

indicates that the produced portion must be one, attempts to prove it, but 

unsuccessfully. Both he and Simplicius practically 

use the same argument. Suppose, says Proclus, 

that the straight lines AC, AD have .« as 1 

common segment With centre B and radius BA 

describe a circle (Post. 3) meeting AC, AD in 

C, D.- Then, since ABC is a straight line through 

the centre, AEC is a semi-circle. Similarly, ABD 

being a straight line through the centre, A ED is a 

semi-circle. Therefore AEC is equal to AED : 

which is impossible. 

Proclus observes that Zeno would object to this proof as really depending 
on the assumption that "two circumferences (of circles) cannot have one 
portion common " ; for this, he would say, is assumed in the common proof 
by superposition of the fact that a circle is bisected by a diameter, since that 
proof takes it for granted that, if one part of the circumference cut off by the 
diameter, when applied to the other, does not coincide with it, it must neces- 
sarily fall either entirely outside or entirely inside it, whereas there is nothing 
to prevent their coinciding, not altogether, but in part only ; and, until you 
really prove the bisection of a circle by its diameter, the above proof is not 
valid. Posidonius is represented as having derided Zeno for not seeing that 
the proof of the bisection of a circle by its diameter goes on just as well if the 
circumferences fail to coincide in fart only. But the true objection to the 
proof above given is that the proof of the bisection of a circle by any diameter 
itself assumes that two straight lines cannot have a common segment; for, if 
we wish to draw the diameter of a circle which has its extremity at a given point 
of the circumference we have to join the latter point to the centre (Post. 1) and 
then to produce the straight line so drawn till it meets the circle again (Post, a), 
and it is necessary for the proof that the produced part shall be unique. 

Saccheri adopted the proper order when he gave, first the proposition that 
two straight lines cannot have a common segment, and after that the 
proposition that any diameter of a circle bisects the circle and its circumference. 

Saccheri's proof of the former is very interesting as showing the thorough- 
ness of his method, if not at the end entirely convincing. It is in five stages 
which I shall indicate shortly, giving the full argument of the first only. 

Suppose, if possible, that AX is a common segment of both the straight 
lines AXB, AXC, in one plane, produced beyond 
X. Then describe about X as centre, with radius 
XB or XC, the arc BMC, and draw through X to 
any point on it the straight line XM. 

(i) I maintain that, with the assumption 
made, the lint AXM is also a straight line which 
is drawn from the point A to the point X and pro- 
duced beyond X. 

For, if this line were not straight, we could draw 
another straight line AM which for its part would 
be straight. This straight line will either (a) cut one 
of the two straight lines XB, XC in a certain point 
A" or (6) enclose one of them, for instance XB, in 
the area bounded by AX, XM and APLM. 




i 9 8 BOOK I [i. Post. 2 

But the first alternative (a) obviously contradicts the foregoing lemma [that 
two straight lines cannot enclose a space], since in that case the two lines 
AXK, ATK, which by hypothesis are straight, would enclose a space. 

The second possibility (b) is at once seen to involve a similar absurdity. 
For the straight Jine X B must, when produced beyond 3, ultimately meet 
APLM in a point L. Consequently the two lines AXBL, APL, which by 
hypothesis are straight, would again enclose a space. If however we were to 
assume that the straight line XB produced beyond B will ultimately meet 
either the straight line XM or the straight line XA in another point, we should 
in the same way arrive at a contradiction. 

From this it obviously follows that, on the assumption made, the line 
AXM is itself the straight line which was drawn from the point A to the point 
M\ and that is what was maintained. 

The remaining stages are in substance these. 

(ii) If the straight line AXB, regarded as rigid, revolves about AX as axis, 
it cannot assume two more positions in the same plane, so that, for example, in 
one position XB should coincide with XC, and in the other -with XM, 

[This is proved by considerations of symmetry. AXB cannot be altogether 
" similar or equal to " AXC, if viewed from the same side {left or right) of 
both : otherwise they would coincide, which by hypothesis they do not. But 
there is nothing to prevent AXB viewed from one side (say the left) being 
" similar or equal to " AXC viewed from the other side (i.e. the right), so that 
AXB can, without any change, be brought into the position AXC. 

AXB cannot however take the position of the other straight line AXM&& 
well- If they were like on one side, they would coincide ; if they were like on 
opposite sides, AXM, AXC would be like on the same side and therefore 
coincide.] 

(iii) The other positions of AXB during the revolution must be above or 
below the original plane. 

(iv) It is next maintained that there is a point D on the are BC such that, if 
XD is drawn, AXD is not only a straight line but is such that viewed from the left 
side it is exactly "similar or equal" to what it is when vieioedfrom the right side. 

[First, it is proved that points M, F can be found on the arc, corresponding 
in the same way as B, C do, but nearer together, and of course AXM, AXF 
are both straight lines. 

Secondly, similar corresponding points can be found still nearer together, 
and so on continually, until either (a) we come to one point D such that AXD 
is exactly like itself when the right and left sides are compared, or (b) there are 
two ultimate points of this sort M, F, so that both AXM, AXF have this 
property. 

Thirdly, {&) is ruled out by reference to the definition of a straight line. 

Hence (a) only is true, and there is only one point D such as described.] 

(v) Lastly, Saccheri concludes that the straight line AXD so determined 
" is alone a straight line, and the immediate prolongation from A beyond X to 
D," relying again on the definition of a straight line as " lying evenly.'' 

Simson deduced the proposition that two straight lines cannot have a 
common segment as a corollary from 1, 11; but his argument is a complete 
petitio principii, as shown by Tod hunter in his note on that proposition. 

Proclus (p. 317, 10) records an ancient proof also based on the proposition 
1. 11. Zeno, he says, propounded this proof and then criticised it. 




t. Post, i, 3] NOTES ON POSTULATES 2, 3 i0o 

Suppose that two straight tines AC, AD have a common segment AB, and 
let BE be drawn at right angles to AC. 

Then the angle EBC is right. 

If then the angle EBD is also right, the two 
angles will be equal ; which is impossible. 

If the angle EBD is not right, draw BE at right 
angles to AD ; therefore the angle FBA is right. 

But the angle EBA is right. 

Therefore the angles EBA, FBA are equal : 
which is impossible. 

Zeno objected to this, says Proclus, because it assumed the later pro- 
position 1. 1 1 for its proof. Posidonius said that there was no trace of such 
a proof to be found in the text-books of Elements, and that it was only invented 
by Zeno for the purpose of slandering contemporary geometers. Posidonius 
maintains further that even this proof has something to be said for it. There 
must be some straight line at right angles to each of the two straight lines A C, 
AD (the very definition of right angles assumes this) : "suppose then it happens 
to be the straight line we have set up." Here then we have an ancient instance 
of a defence of hypothetical construction, but in such apologetic terms (" it is 
possible to say something even for this proof") that we may conclude that in 
general it would not have been accepted by geometers of that time as a 
legitimate means of proving a proposition. 

Todhunter proposed to deduce that ftvo straight lines cannot have a 
common segment from i. 13. But this will not serve either, since, as before 
mentioned, the assumption is really required for i. 4. 

It is best to make it a postulate. 

Postulate 3. 

Kol itavTi Wirpcii ttal 8ta<mj/iaTL kvuXov ypafaa&at. 

To describe a circle with any centre and distance. 

In this case Euclid's text has the passive of the verb : "a circle can be 
drawn " ; Proclus however has the active {yputyu) as Euclid has in the first 
two Postulates. 

Distance, Swo-nJfiaTt. This word, meaning '' distance " quite generally (cf. 
Arist. Metaph. 1055 a 9 "it is between extremities that distance is greatest," 
ibid. 1056 a 36 " things which have something between them, that is, a certain 
distance "), and also " distance " in the sense of " dimension " (as in " space 
has three dimensions, length, breadth and depth," Arist. Physics iv. 1, 209 a 4), 
was the regular word used for describing a circle with a certain radius, the 
idea being that each point of the circumference was at that distance from the 
centre (cf. Arist Afetearologica ill. 5, 376 b 8 : "if a circle be drawn... with 
distance Mil "). The Greeks had no word corresponding to radius : if they 
had to express it, they said "(straight lines) drawn from the centre" («u « tou 
nfrrpov, Eucl. lit. Def. 1 and Prop. 26 \ Mtteorologica 11.5,362 b! has the full 
phrase oi in toS kot^ob ayo/MW" ypauftai). 

Mr Frankland observes that it would be remarkable if, unlike Postulates 1 
and 2, this Postulate implied merely what tt says, that a circle can be drawn 
with any centre and distance. We may regard it, if we please, as helping to the 
complete delineation of the Space which Euclid's geometry is to investigate 
formally. The Postulate has the effect of removing any restriction upon the 
siie of the circle. It may (1) be indefinitely small, and this implies that space 
is continuous, not discrete, with an irreducible minimum distance between 



*oo BOOK I [i. Post 3. 4 

contiguous points in it (a) The circle may be indefinitely large, which 
implies the fundamental hypothesis of infinitude of space. This last assumed 
characteristic of space is essential to the proof of 1. 16, a theorem not 
universally valid in a space which is unbounded in extent but finite in size. It 
would however be unsafe to suppose that Euclid foresaw the use to which his 
Postulate might thus be put, or formulated it with such an intention. 

Postulate 4. 

That all fight angles are equal to one another. 

While this Postulate asserts the essential truth that a right angle is a 
determinate magnitude so that it really serves as an invariable standard by 
which other (acute and obtuse) angles may be measured, much more than 
this is implied, as will easily be seen from the following consideration. If the 
statement is to be proved, it can only be proved by the method of applying one 
pair of right angles to another and so arguing their equality. But this method 
would not be valid unless on the assumption of the invariability of figures, 
which would therefore have to be asserted as an antecedent postulate. Euclid 
preferred to assert as a postulate, directly, the fact that all right angles are 
equal ; and hence his postulate must be taken as equivalent to the principle of 
invariability of figures or its equivalent, the homogeneity of space. 

According to Proclus, Geminus held that this Postulate should not be 
classed as a postulate but as an axiom, since it does not, like the first three 
Postulates, assert the possibility of some construction but expresses an essential 
property of right angles. Proclus further observes (p. i88, 8} that it is not a 
postulate in Aristotle's sense either. (In this I think he is wrong, as explained 
above.) Proclus himself, while regarding the assumption as axiomatic ("the 
equality of right angles suggests itself even hy virtue of our common notions"), 
is prepared with a proof, if such is asked for. 

Let ABC, DEF be two right 
angles. 

If they are not equal, one of them 
must be the greater, say ABC. 

Then, if we apply DE to AB, EF 
will fall within ABC, as BG. 

Produce CB to H. Then, since 
ABC is a right angle, so is ABU, and the two angles are equal (a right angle 
being by definition equal to its adjacent angle). 

Therefore the angle ABH is greater than the angle ABG. 

Producing GB to K, we have similarly the two angles ABK, ABG both 
right and equal to one another; whence the angle ABH is less than the angle 
ABG. 

But it is also greater : which is impossible. 

Therefore etc. 

A defect in this proof is the assumption that CB t GB can each be 
produced only in one way, and that BK falls outside the angle ABH. 

Saccheri's proof is more careful in that he premises a third lemma in 
addition to those asserting <i) that two straight lines 
cannot enclose a space and (*) that two straight lines 
cannot have a common segment. The third lemma is : 
If two straight lines AB, CXD meet one another at an 
intermediate point X, they do not touch at that point, but 
cut one another. 





i. Post. 4] 



NOTES ON POSTULATES 3, 4 



*oi 




M 



Suppose now that DA standing on BAC makes the two angles DAB, 
DAC equal, so that each is a right angle by the definition. 

Similarly, let LHioTm. with the straight line FHM the right angles LHF, 
LHM. 

Let DA, HL be equal ; and sup- 
pose the whole of the second figure 
so laid upon the first that the point 
H falls on A, and L on D. 

Then the straight line FHM 'will 
(by the third lemma) not touch the 
straight line BC at A ; it will either 

(a) coincide exactly with BC, or 

(0) cut it so that one of its extremities, as F, will fall above [BC] and the 
other, M, below it. 

If the alternative (a) is true, we have already proved the exact equality of 
all rectilineal right angles. 

Under alternative (e) we prove that the angle I.HF, being equal to the 
angle DAF, is less than the angle DAB or DAC, and a fortiori less than the 
angle DAM ox LHM; which is contrary to the hypothesis. 

[Hence (a) is the only possible alternative, so that all right angles are 
equal.] 

Saicheri adds that it makes no difference if the. angle DAF diverges 
infinitely lift It from the angle DAB. This would equally lead to a conclusion 
contradicting the hypothesis. 

It will be observed that Saccheri speaks of "the exact equality of all 
rectilineal right angles." He may have had in mind the remark of Pappus, 
quoted by Proclus (p. t8a, 1 1), that the converse of 
this postulate, namely that an angle which is equal 
to a right angle is also right, is not necessarily true, 
unless the former angle is rectilineal. Suppose two 
equal straight lines BA, BCsX right angles to one 
another, and semi-circles described on BA, BC 
respectively as AEB, BDC in the figure. Then, 
since the semi-circles are equal, they coincide .if 
applied to one another. Hence the "angles" 
BBA, DBC are equal. Add to each the " angle " 
ABD ; and it follows that the iunular angle EBD is equal to the right angle 
ABC. (Similarly, if BA, BCbt inclined at an acute or obtuse angle, instead 
of at a right angle, we find a Iunular angle equal to an acute or obtuse angle.) 
This is one of the curiosities which Greek commentators delighted in. 

Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all 
right angles are equal from the equivalent fact that all flat angles are equal, 
which is either itself assumed as a postulate or immediately deduced from some 
other postulate. 

Hilbert takes quite a different line. He considers that Euclid did wrong 
in placing Post 4 among "axioms." He himself, after his Group in. of 
Axioms containing six relating to congruence, proves several theorems about 
the congruence of triangles and angles, and then deduces our Postulate. 

As to the raison fHre and the place of Post. 4 one thing is quite certain. 
It was essential from Euclid's point of view that it should come before Post. 5, 
since the condition in the latter that a certain pair of angles are together less 
than two right angles would be useless unless it were first made clear that 
right angles are angles of determinate and invariable magnitude. 




jo* BOOK I [i. Post. 5 



Postulate 5. 

K HI cav cfc Sue tv@twS t\$<ltt fynriVrovtra T4V ^VTOf Kcii tVt TO. ctvrii ft* 1*7) ytavia s 
Silo upCuS* Aa<r<rovas iroip, (u^aAAo/itfas To! Sw> tWtiM i»' aiMipw <ri>/ijrtnTMif, 
c£ a ,r< i,ij^ (tcrtv ai toJv Buo op&uv eX<£crf/in'€!. 

TXrt/, if a straight line falling on two straight lints make the interior angles 
on the same side less than two right angles, the two straight lines, if produeed 
indefinitely, meet on that side on whieh are the angles less than the tit<o right 
angles. 

Although Aristotle gives a clear idea of what he understood by a postulate, 
he does not give any instances from geometry; still less has he any allusion 
recalling the particular postulates found in Euclid. We naturally infer that 
the formulation of these postulates was Euclid's own work. There is a more 
positive indication of the originality of Postulate 5, since in the passage {A rial 
prior. 11. 16, 65 a 4) quoted above in the note on the definition of parallels he 
alludes to some petitio principii involved in the theory of parallels 'current in 
his time. This reproach was removed by Euclid when he laid down this 
epoch-making Postulate. When we consider the countless successive attempts 
made through more than twenty centuries to prove the Postulate, many of 
them by geometers of ability, we cannot but admire the genius of the man 
who concluded that such a hypothesis, which he found necessary to the 
validity of his whole system of geometry, was really indemonstrable. 

From the very beginning, as we know from Proclus, the Postulate was 
attacked as such, and attempts were made to prove it as a theorem or to get 
rid of it by adopting some other definition of parallels; while in modem times 
the literature of the subject is enormous. Riccardi (Saggio di una bibliografia 
Euclidea, Part iv,, Bologna, 1890) has twenty quarto pages of titles of mono- 
graphs relating to Post. 5 between the dates 1607 and 1887. Max Simon 
{Ueber die Entwickit/ng der Elementar-geometrie im XIX. Jakrhundert, 1906) 
notes that he has seen three new attempts, as late as 1891 (a century after 
Gauss laid the foundation of non-Euclidean geometry), to prove the theory of 
parallels independently of the Postulate. Max Simon himself (pp. 53 — 61) 
gives a large number of references to books or articles on the subject and 
refers to the copious information, as to contents as well as names, con- 
tained in Schotten's Inhalt mid Methode des planimetrisehen Unterrichts, 11. 

PP '83— 33 2 - 

This note will include some account of or allusion to a few of the most 
noteworthy attempts to prove the Postulate. Only those of ancient times, as 
being less generally accessible, will be described at any length ; shorter 
references must suffice in the case of the modern geometers who have made 
the most important contributions to the discussion of the Postulate and have 
thereby, in particular, contributed most towards the foundation of the non- 
Euclidean geometries, and here I shall make use principally of the valuable 
Article 8, Sulla teoria delle paraliele e suite geometric non-euclidee (by Roberto 
Bonola), in Quesiioni riguardanti le matematiehe elementari, I. pp. 147 — 363. 

Proclus (p. 191, xi sqq.) states very clearly the nature of the first objec- 
tions taken to the Postulate. 

" This ought even to be struck out of the Postulates altogether ; for it is a 
theorem involving many difficulties, which Ptolemy, in a certain book, set 
himself to solve, and it requires for the demonstration of it a number 
of definitions as well as theorems. And the converse of it is actually 
proved by Euclid himself as a theorem. It may be that some would be 



t. Post. 5] NOTE ON POSTULATE 5 203 

deceived and would think it proper to place even the assumption in question 
among the postulates as affording, in the lessening of the two right angles, 
ground for an instantaneous belief that the straight lines converge and meet. 
To such as these Geminus correctly replied that we have learned from the 
very pioneers of this science not to have any regard to mere plausible imagin- 
ings when it is a question of the reasonings to be included in our geometrical 
doctrine. For Aristotle says that it is as justifiable Co ask scientific proofs of 
a rhetorician as to accept mere plausibilities from a geometer; and Sim mi as is 
made by Plato to say that he recognises as quacks those who fashion for 
themselves proofs from probabilities. So in this case the fact that, when the 
right angles are lessened, the straight lines converge is true and necessary; 
but the statement that, since they converge more and more as they are pro- 
duced, they will sometime meet is plausible but not necessary, in the absence 
of some argument showing that this is true in the case of straight tines. For 
the fact that some lines exist which approach indefinitely, but yet remain 
non-secant (itrvpirrtirroi), although it seems improbable and paradoxical, is 
nevertheless true and fully ascertained with regard to other species of lines. 
May not then the same thing be possible in the case of straight lines which 
happens in the case of the lines referred to ? Indeed, until the statement in 
the Postulate is clinched by proof, the facts shown in the case of other lines 
may direct our imagination the opposite way. And, though the controversial 
arguments against the meeting of the straight lines should contain much that 
is surprising, is there not all the more reason why we should expel from our 
body of doctrine this merely plausible and unreasoned (hypothesis) ? 

"It is then clear from this that we must seek a proof of the present 
theorem, and that it is alien to the special character of postulates. But how 
it should be proved, and by what sort of arguments the objections taken to 
it should be removed, we must explain at the point where the writer of the 
Elements is actually about to recall it and use it as obvious. It will be 
necessary at that stage to show that its obvious character does not appear 
independently of proof, but is turned by proof into matter of knowledge." 

Before passing to the attempts of Ptolemy and Proclus to prove the 
Postulate, I should note here that Simplicius says (in an-Nairlzi, ed. Besthorn- 
Heiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest, 
but requires proof, and accordingly " Abthiniathus " and Diodorus had 
already proved it by means of many different propositions, while Ptolemy also 
had explained and proved it, using for the purpose Eucl. t. 13, 15 and 16 (or 
18). The Diodorus here mentioned may be the author of the Analtmma on 
which Pappus wrote a commentary. It is difficult even to frame a conjecture 
as to who "Abthiniathus" is. In one place in the Arabic text the name 
appears to be written " Anthisathus " (H. Suter in Zeztstkrifi fiir Math, und 
Phystk, xxxviii,, hist. litt. Abth. p. 194). It has occurred to me whether he 
might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was 
long known as Serenus of Antissa. Serenus says (De sittione cyiindri, ed. 
Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his, 
was not satisfied with what Euclid said, but showed their nature more cleverly 
by an example; for he says that parallel straight lines are such a thing as we 
see on walls or on the ground in the shadows of pillars which are made when 
either a torch or a lamp is burning behind them. And, although this has only 
been matter of merriment to every one, I at least must not deride it, for the 
respect I have for the author, who is my friend." If Peithon was known as 
" of Antinoeia " or " of Antissa," the two forms of the mysterious name might 
perhaps be an attempt at an equivalent ; but this is no more than a guess. 



so 4 BOOK I [i. Post. 5 

Simplicity adds in full and word for word the attempt of his " friend " or 
his " master Aganis " to prove the Postulate. 

Proclus returns to the subject (p, 365, 5) in his note on Eucl. 1. 29. He 
says that before his time a certain number of geometers had classed as a 
theorem this Euclidean postulate and thought it matter for proof, and he then 
proceeds to give an account of Ptolemy's argument. 

Noteworthy attempts to prove the Postulate. 

Ptolemy. 

We learn from Proclus (p. 365, 7 — -11) that Ptolemy wrote a book on the 
proposition that " straight lines drawn from angles less than two right angles 
meet if produced," and that he used in his "proof" many of the theorems in 
Euclid preceding 1. 19. Proclus excuses himself from reproducing the early 
part of Ptolemy's argument, only mentioning as one of the propositions 
proved in it the theorem of Eucl. 1. 28 that, if two straight lines meeting a 
transversal make the two interior angles on the same side equal to two right 
angles, the straight lines do not meet, however far produced. 

I. From Proclus' note on I. 28 (p. 362, 14 sq.) we know that Ptolemy 
proved this somewhat as follows. 

Suppose that there are two straight lines A B, • CD, and that EFGff, 
meeting them, makes the angles BFG, FGD equal to two right angles. 
I say that AB, CD are parallel, that is, they 
are non-secant. 

For, if possible, let FB, GD meet at K.' 

Now, since the angles BFG, FGD are 
equal to two right angles, while the four 
angles AFG, BFG, FGD, FGC are together 
equal to four right angles, 

the angles AFG, FGC are equal to two 
right angles. 

"If therefore FB, GD, when the interior angles are equal to two right 
angles, meet at K, the straight lines FA, GC will also meet if produced ',- for the 
angles AFG, CGFare also equal to two right angles. 

"Therefore the straight lines will either meet in both directions or in 
neither direction, if the two pairs of interior angles are both equal to two right 
angles. 

" Let, then, FA, GC meet at L. 

"Therefore the straight lines LABK, LCDK enclose a space; which is 
impossible. 

" Therefore it is not possible for two straight lines to meet when the 
interior angles are equal to two right angles. Therefore they are parallel." 

[The argument in the words italicised would be clearer if it" had been 
shown that the two interior angles on one side of EH are severally equal to the 
two interior angles on the other, namely BFG to CGF and FGD to AFG ; 
whence, assuming FB, GD to meet in K, we can take the triangle KFG and 
place it (e.g. by rotating it in the plane about O the middle point of FG) so 
that FG falls where GFis in the figure and GD falls on FA, in which case 
FB must also fall on GC; hence, since FB, GD meet at K, GC and FA 
must meet at a corresponding point L, Or, as Mr Frank land does, we may 
substitute for FG a straight line MN through the middle point of FG 
drawn perpendicular to one of the parallels, say AB. Then, since the two 
triangles OMF, ONG have two angles equal respectively, namely FOM to 




F 

7 



i. Post. 5] NOTE ON POSTULATE 5 105 

GON(i. 15) and OFM to 0GJV; and one side OF equal to one side OG, the 
triangles are congruent, the angle ONG is a right angle, and MN is perpen- 
dicular to both AB and CD. Then, by the same method of application, 
MA, NC are shown to form with MN a triangle MALCN congruent with 
the triangle NDKBM, and MA, NC meet at a point L corresponding to K. 
Thus the two straight lines would meet at the two points K, L, This is what 
happens under the Riemann hypothesis, where the axiom that two straight 
lines cannot enclose a space does not hold, but all straight lines meeting in 
one point have another point common also, and e,g. in the particular figure 
just used K, L are points common to all perpendiculars to MN. If we 
suppose that K, L are not distinct points, but one point, the axiom that two 
straight lines cannot enclose a space is not contradicted.] 

II. Ptolemy now tries to prove 1. 29 without using our Postulate, and 
then deduces the Postulate from it (Proclus, pp. 365, 14 — 367, 37). 

The argument to prove 1. 29 is as follows. 

The straight line which cuts the parallels must make the sum of the 
interior angles on the same side equal to, greater 
than, or less than, two right angles. b £ P 

"Let AB, CD be parallel, and let FG meet 
them. I say (1) that FG does not make the 
interior angles on the same side greater than two £— 
right angles. 

" For, if the angles AFG, CGF are greater than two right angles, the 
remaining angles BFG, DGF are less than two right angles. 

" But the same two angles are also greater than two right angles ; for AF, 
CG are no more parallel than FB, GD, so that, if the straight line falling on 
AF, CG makes the interior angles greater than two right angles, the straight line 
falling on FB, GD will also make the interior angles greater than two right 
angles. 

" But the same angles are also less than two right angles ; for the four 
angles AFG, CGF, BFG, DGF are equal to four right angles : 
which fa impossible 

*' Similarly (2) we can show that the straight line falling on the parallels 
does not make the interior angles on the same side less than two right angles. 

" But (3), if it makes them neither greater nor less than two right angles, 
it can only make the interior angles on the same side equal to two right 
angles." 

III, Ptolemy deduces Post 5 thus : 

Suppose that the straight lines making angles with a transversa) less than 
two right angles do not meet on the side on which those angles are. 

Then, a fortiori, they will not meet on the other side on which are the 
angles greater than two right angles. 

Hence /he straight lines will not meet in either direction ; they are there- 
fore parallel. 

But, if so, the angles made by them with the transversal are equal to two 
right angles, by the preceding proposition (= 1. 29). 

Therefore the same angles will be both equal to and less than two right 
angles: 
which is impossible. 

Hence the straight lines will meet. 




ao6 BOOK I [i. Post. 5 

IV. Ptolemy lastly enforces his conclusion that the straight lines will 
meet on the side on which are the angles less than two right angles by recurring 
to the a fortiori step in the foregoing proof. 

Let the angles AFG, CGF in the accompanying figure be together less 
than two right angles. 

Therefore the angles BFG, DGF are greater 
than two right angles. 

We have proved that the straight lines are not 
non-secant. 

If they meet, they must meet either towards 
A, C, or towards B, D. 

(i) Suppose they meet towards B, £>, at K. 

Then, since the angles AFG, CGFsze. less than 
two right angles, and the angles AFG, GFB are 
equal to two right angles, take away the common angle AFG, and 

the angle CGF is less than the angle BFG; 

that is, the exterior angle of the triangle KFG is less than the interior arid 
opposite angle BFG : 
which is impossible. 

Therefore AB, CD do not meet towards B, D. 

(v) But they do meet, and therefore they must meet in one direction or 
the other: 

therefore they meet towards A, B, that is, on the side where are the 
angles less than two right angles. 

The flaw in Ptolemy's argument is of course in the part of his proof of 
i. 2 9 which I have italicised. As Proclus says, he is not entitled to assume 
that, if AB, CD are parallel, whatever is true of the interior angles on one 
side of FG (i.e. that they are together equal to, greater than, or less than, two 
right angles} is necessarily true at the same time of the interior angles on the 
other side. Ptolemy justifies this by saying that FA, GC are no more parallel 
in one direction than FB, GD are in the other : which is equivalent to the 
assumption that through any point only one parallel can be drawn to a given 
straight line. That is, he assumes an equivalent of the very Postulate he is 
endeavouring to prove. 

Proclus. 

Before passing to his own attempt at a proof, Proclus (p. 368, 26 sqq.) 
examines an ingenious argument (recalling somewhat the famous one about 
Achilles and the tortoise) which appeared to show that it was impossible for 
the lines described in the Postulate to meet. 

Let AB, CD make with AC the angles BAC, ACD together less than 
two right angles. 

Bisect AC a.t E and along AB, CD 
respectively measure AF, CG so that each 
is equal to AE. gl \ |h 

Bisect FG at K and mark off FK, 
GL each equal to FH; and so on. 

Then AF, CG will not meet at any 
point on FG ; for, if that were the case, two sides of a triangle would be 
together equal to the third : which is impossible. 



1. Post. 5] NOTE ON POSTULATE 5 *>7 

Similarly, AB, CD will not meet at any point on KL; and "proceeding 
like this indefinitely, joining the non-coincident points, bisecting the lines so 
drawn, and cutting off from the straight lines portions equal to the half of 
these, they say they thereby prove that the straight lines AB, CD will not 
meet anywhere." 

It is not surprising that Proclus does not succeed in exposing the fallacy 
here (the fact being thai the process will indeed be endless, and yet the straight 
lines will intersect within a finite distance). But Proclus' criticism contains 
nevertheless something of value. He says that the argument will prove too 
much, since we have only to join A G in order to see that straight lines making 
svme angles which are together less than two right angles do in fact meet, 
namely AG, CG. "Therefore it is not possible to assert, without some definite 
limitation, that the straight lines produced from angles less than two right 
angles do not meet. On the contrary, it is manifest that MM straight lines, 
when produced from angles less than two right angles, do meet, although the 
argument seems to require it to be proved that this property belongs to all 
such straight lines. For one might say that, the lessening of the two right 
angles being subject to no limitation, with such and such an amount of 
lessening the straight tines remain non-secant, but with an amount of lessening 
in excess of this they meet (p. 371, 2 — 10)." 

[Here then we have the germ of such an idea as that worked out by 
Lobachewsky, namely that the straight lines issuing from a point in a plane 
can be divided with reference to a straight line lying in that plane into two 
classes, "secant" and "non-secant," and that we may define as parallel the 
two straight lines which divide the secant from the non-secant class.] 

Proclus goes on (p. 371, io) to base his own argument upon "an axiom 
such as Aristotle too used in arguing that the universe is finite. For, if from 
one point two straight lines forming an angle be produced indefinitely, the distance 
(StaiTTairit, Arist. htaimjjta) between the said straight tines produced indefinitely 
will exceed any finite magnitude. Aristotle at all events showed that, if the 
Straight lines drawn from the centre to the circumference are infinite, the 
interval between them is infinite. For, if it is finite, it is impossible to 
increase the distance, so that the straight lines {the radii) are not infinite. 
Hence the straight lines, when produced indefinitely, will be at a distance from 
one another greater than any assumed finite magnitude." 

This is a fair representation of Aristotle's argument in De caelo ]. 5, 271 
b 28, although of course it is not a proof of what Proclus assumes as an 
axiom. 

This being premised, Proclus proceeds (p. 371, 24): 

I. " I say that, if any straight line cuts one of two parallels, it will cut 
the other also. 

"For let AB, CD be parallel, and let EFG cut AB; I say that it will cut 
CD also. 

" For, since BF, FG are two straight lines from .E 

one point F, they have, when produced indefinitely, ft X^ B 

a distance greater than any magnitude, so that it will ^\ 

also be greater than the interval between the parallels. " 

Whenever therefore they are at a distance from one 

another greater than the distance between the parallels, 
FG will cut CD. 

" Therefore etc." 




2o8 BOOK I [i. Post. 5 

II. " Having proved this, we shall prove, as a deduction from it, the 
theorem in question. 

"For let AB, CD be two straight lines, and let EF falling on them make 
the angles BEF, DFE less than two right angles. 

"I say that the straight lines will meet on that 
side on which are the angles less than two right 
angles. 

" For, since the angles BEF, DFE are less 
than two right angles, let the angle HEB be equal 
to the excess of two right angles (over them), and let HE be produced to K. 

" Since then EF falls on KH, CD and makes the two interior angles 
HEF, DFE equal to two right angles, 

the straight lines MX, CD are parallel. 

"And AB cuts KH\ therefore it will also cut CD, by what was before 
shown. 

" Therefore AB, CD will meet on that side on which are the angles less 
than two right angles. 

" Hence the theorem is proved." 

Clavius criticised this proof on the ground that the axiom from which 
it starts, taken from Aristotle, itself requires proof. He points out that, just 
as you cannot assume that two lines which continually approach one another 
will meet (witness the hyperbola and its asymptote), so you cannot assume 
that two lines which continually diverge will ultimately be so far apart that a 
perpendicular from a point on one let fall on the other will be greater than 
any assigned distance ; and he refers to the conchoid of Nkomedes, which 
continually approaches its asymptote, and therefore continually gets farther 
away from the tangent at the vertex • yet the perpendicular from any point on 
the curve to that tangent will always be less than the distance between the 
tangent and the asymptote. Saccheri supports the objection. 

Proclus' first proposition is open to the objection that it assumes that two 
"parallels" (in the Euclidean sense) or, as we may say, two straight liius 
which have a common perpendicular, are (not necessarily equidistant, but) 
so related that, when they are produced indefinitely, the perpendicular from a 
point of one upon the other remains finite. 

This last assumption is incorrect on the hyperbolic hypothesis ; the 
"axiom" taken from Aristotle does not hold on the elliptic hypothesis, 

Nasiraddln at.-TusI. 

The Persian-bom editor of Euclid, whose date is 1201— 1274, has three 
lemmas leading up to the final proposition. Their content is substantially as 
follows, the first lemma being apparently assumed as evident 

I. (o) If AB, CD be two straight lines such that successive perpen- 
diculars, as EF, GH, KL, from points on AB to CD always make with A3 
unequal angles, which are always acute on the side towards B and always 
obtuse on the side towards A, then the lines AB, 
CD, so long as they do not cut, approach continually 
nearer in the direction of the acute angles and diverge 
continually in the direction of the obtuse angles, and 
the perpendiculars diminish towards B, D, and in- 
crease towards A, C. 

(6) Conversely, if the perpendiculars so drawn 



L H F C 

continually become shorter in the direction of B, D, and longer in the 



i. Post. 3] 



NOTE ON POSTULATE 5 



zog 




direction of A, C, the straight lines AB, CD approach continually nearer in 
the direction of B, D and diverge continually in the other direction ; also 
each perpendicular will make with AB two angles one of which is acute and 
the other is obtuse, and all the acute angles will lie in the direction towards 
B, D, and the obtuse angles in the opposite direction. 

[Saccheri points out that even the first part {a) requires proof. As 
regards the converse (b) he asks, why should not the successive acute angles 
made by the perpendiculars with AB, while remaining acute, become greater 
and greater as the perpendiculars become smaller until we arrive at last at a 
perpendicular which is a common perpendicular to both lines? If that happens, 
alt the author's efforts are in vain. And, if you are to assume the truth of the 
statement in the lemma without proof, would it not, as Wall is said, be as 
easy to assume as axiomatic the statement in Post. 5 without more ado?] 

II. ^AC, BI) be drawn from the extremities of AB at right angles to it 
and on the same side, and if AC, EDfc made equal to one another and CD be 
joined, each of the angles ACD, BDC will be right, and 
CD will be equal to AB, 

The first part of this lemma is proved by redwtio ad 
absurdum from the preceding lemma. If, e.g., the angle 
A CD is not right, it must either be acute or obtuse. 

Suppose it is acute ; then, by lemma 1, A C is greater 
than BD, which '\s contrary to the hypothesis. And so on. 

The angles ACD, BDC being proved to be right angles, it is easy to 
prove that AB, CD are equal. 

[It is of course assumed in this " proof" that, if the angle ACD is acute, 
the angle BDC is obtuse, and vice versa.] 

III. /// any triangle the three angles are together equal to two right angles. 
This is proved for a right-angled triangle by means of the foregoing lemma, 

the four angles of the quadrilateral ABCD of that lemma being all right angles. 
The proposition is then true for any triangle, since any triangle can be divided 
into two right-angled triangles 

IV. Here we have the final " proof " of Post. 5. Three cases are 
distinguished, 'but it is enough to show the case where one of the interior 
angles is right and the other acute. 

Suppose AB, CD to be two straight lines met by FCE making the angle 
ECD a right angle and the angle CEB 
an acute angle. 

'lake any point G on EB, and draw 
GH perpendicular to EC. 

Since the angle CEG is acute, the 
perpendicular GH will fall on the side of 
E towards D, and will either coincide 
with CD or not coincide with it. In the 
former case the proposition is proved. 

If GH does not coincide with CD 
but falls on the side of it towards F, CD, being within the triangle formed by 
the perpendicular and by CE, EG, must cut EG. [An axiom is here used, 
namely that, if CD be produced far enough, it must pass outside the triangle 
and therefore cut some side, which must be EB, since it cannot be the 
perpendicular (1. 27), or CE.) 

Lasdy, let C.fffall on the side of CD towards E. 




aro BOOK I [i. Post. 5 

Along HC set off HK, KL etc., each equal to EH, until we get the first 
point of division, as M, beyond C. 

Along GB set off GN, NO etc, each equal to EG, until EP is the same 
multiple of EG that EM is of EH. 

Then we can prove that the perpendiculars from N, 0, P on EC fall on 
the points K, L, M respectively. 

For take the first perpendicular, that from N, and call it NS. 

Draw EQaa right angles to EH and equal to GH, and set off SR along 
SiValso equal to GH, Join QG, GR, 

Then (second lemma) the angles EQG, QGHaie right, and QG = EH. 

Similarly the angles SRG, RGHwe right, and RG-SH 

Thus RGQ is one straight line, and the vertically opposite angles NGR, 
EGQ are equal. The angles NRG, EQG are both right, and NG = GE, by 
construction. 

Therefore (1. 26") EG =■ GQ ; 

whence SH= HE = KH, and S coincides with K. 

We may proceed similarly with the other perpendiculars. 

Thus PM is perpendicular to EE. Hence CD, being parallel to MP and 
within the triangle PME, must cut EP, if produced far enough. 

John Wallis. 

As is well known, the argument of Wallis (1616—1703) assumed as a 
postulate that, given a figure, another figure is possible which is similar to (he 
given one and of any sine whatever. In fact Wallis assumed this for triangles 
only. He first proved (1) that, if a finite straight line is placed on an infinite 
straight line, and is then moved in its own direction as far as we please, 
it will always lie on the same infinite straight line, (2) that, if an angle be 
moved so that one leg always slides along an infinite straight line, the angle 
will remain the same, or equal, (3) that, if two straight lines, cut by a third, 
make the interior angles on the same side less than two right angles, each 
of the exterior angles is greater than the opposite 
interior angle (proved by means of 1. 13). p x 

(4) UAB, CD make, with AC, the interior 
angles less than two right angles, suppose AC 

(with AB rigidly attached to it) to move along _ ^ \- g 

AF to the position ay, such that a coincides 

with C. If AB then takes the position aft, o£ lies entirely outside CD (proved 
by means of {3) above). 

(5) With the same hypotheses, the straight line aft, sr AB, during Us 
motion, and before a reaches C, must cut the straight tine CD. 

S6) Here is enunciated the postulate stated above. 
7-) Postulate S is now proved thus. 

Let AB, CD be the straight lines which make, with the infinite straight 
line ACF meeting them, the interior angles 
BA C, DC A together less than two right angles. 

Suppose AC (with AB rigidly attached to 
it) to move along ACF until AB takes the 
position of aft cutting CD in it. 

Then, «CVr being a triangle, we can, by 
the above postulate, suppose a triangle drawn 
on the base CA similar to the triangle aCVr. 

Let it be ACF. 

[Wallis here interposes a defence of the hypothetical construction.] 




1. Post. 5] NOTE ON POSTULATE 5 sit 

Thus CP and AP meet at P; and, as by the definition of similar figures 
the angles of the triangles PCA, rCa are respectively equal, the angle PCA 
being equal to the angle rCa and the angle PAC to the angle miCor BAC, 
it follows that CP, APMe on CD, A3 produced respectively. 

Hence AB, CD meet on the side on which are the angles less than two 
right angles. 

[The whole gist of this proof lies in the assumed postulate as to the 
existence of similar figures ; and, as Saccheri points out, this is equivalent to 
unconditionally assuming the "hypothesis of the right angle," and consequently 
Euclid's Postulate 5.] 

Gerolamo Saccheri. 

The book Euclides ab omni naevo vindicatus (1733) by GeTolamo Saccheri 
{1667 — 1733), a Jesuit, and professor at the University of Pavia, is now 
accessible (1) edited in German by Engel and Stackel, Die Theorie dtr 
Parallellinien von Euhtid bis auf Gauss, 1895, pp. 41 — 136, and (2) in an 
Italian version, abridged but annotated, L'Euclide emendato del P. Gerolamo 
Saccheri, by G. Boccardini (Hoepli, Milan, 1904}. It is of much greater 
importance than all the earlier attempts to prove Post. 5 because Saccheri 
was the first to contemplate the possibility of hypotheses other than that of 
Euclid, and to work out a number of consequences of those hypotheses. 
He was therefore a true precursor of Legendie and of Lobachewsky, as 
Beltrami called him (1889), and, it might be added, of Riemann also. For, 
as Veronese observes {Fondamenti di geometria, p, 570), Saccheri obtained 
a glimpse of the theory of parallels in all its generality, while Legendre, 
Lobachewsky and G, Bolyai excluded a priori, without knowing it, the "hypo- 
thesis of the obtuse angle," or the Riemann hypothesis. Saccheri, however, 
was the victim of the preconceived notion of his time that the sole possible 
geometry was the Euclidean, and he presents the curious spectacle of a man 
laboriously erecting a structure upon new foundations for the very purpose of 
demolishing it afterwards ; he sought for contradictions in the heart of the 
systems which he constructed, in order to prove thereby the falsity of his 
hypotheses. 

For the purpose of formulating his hypotheses he takes a plane quadri- 
lateral ABDC, two opposite sides of which, A C, BD, 
are equal and perpendicular to a third AB. Then the 
angles at C and D are easily proved to be equal. On 
the Euclidean hypothesis they are both right angles; 
but apart from this hypothesis they might be both 
obtuse or both acute. To the three possibilities, whicfc 
Saccheri distinguishes by the names (1) the hypothesis of 
the right angle, (i) the hypothesis of the obtuse angle and 
(3) the hypothesis of the acute angle respectively, there corresponds a certain 
group of theorems ; and Saccheri's point of view is that the Postulate will 
be completely proved if the consequences which follow from the last two 
hypotheses comprise results inconsistent with one another. 

Among the most important of his propositions are the following : 

(1) If the hypothesis of the right angle, or of the obtuse angle, or of the acute 
angle is proved true in a single case, it is true in every other case. (Props, v., 

VI., VII.) 

(2) According as the hypothesis of the right angle, the obtuse angle, or the 
acute angle is true, the sum of the thru angles of a triangle is equal to, greater 
than, or less than two right angles. (Prop. i\. ) 




lit BOOK I [i. Post. 5 

(3) From the existence of a single triangle in which the sum of the angles is 
equal to, greater than, or less than two right angles (lie truth of the hypothesis 
of the right angle, obtuse angle, or acute angle respectively follows. (Prop, xv.) 

These propositions involve the following : If in a single triangle the sum 
of the angles is equal to, greater than, or less than two right angles, then any 
triangle has the sum of its angles equal to, greater than, or less than tlt'O right 
angles respectively, which was proved about a century later by Legendre for 
the two cases only where the sum is equal to or less than two right angles. 

The proofs are not free from imperfections, as when, in the proofs of 
Prop. xii. and the part of Prop. xm. relating to the hypothesis of the obtuse 
angle, Saccheri uses Eucl. 1. 18 depending on 1. 16, a proposition which is 
only valid on the assumption that straight lines are infinite in length ; for this 
assumption itself does not hold under the hypothesis of the obtuse angle 
(the Riemann hypothesis). 

The hypothesis of the acute angle takes Saccheri much longer to dispose 
of, and this part of the book is less satisfactory ; but it contains the following 
propositions afterwards established anew by Lobachewsky and Bolyai, viz. : 

(4) Two straight lines in a plane {even on the hypothesis of the acute 
angle) either have a common perpendicular, or must, if produced in one and the 
same direction, either intersect once at a finite distance or at least continually 
approach one another. {Prop, xxin.) 

(5) In a cluster of rays issuing from a point there exist always (on the 
hypothesis of the acute angle) two determinate straight lines which separate the 
straight lines which intersect a fixed straight line from those which do not 
intersect it, ending with and including the straight line which has a common 
perpendicular with the fixed straight line. (Props. XXX., xxxc, xxxii.) 

Lambert. 

A dissertation by G.S. Kliigel, Conatuum praecipuorum tlteoriamparallelarnm 
demonstrandi recensio (1 763), contained an examination of some thirty " demon- 
strations" of Post. 5 and is remarkable for its conclusion expressing, apparently 
for the first time, doubt as to its demenstrability and observing that the 
certainty which we have in us of the truth of the Euclidean hypothesis is 
not the result of a series of rigorous deductions but rather of experimental 
observations. It also had the greater merit that it called the attention of 
Johann Heinrich Lambert (1728—1777) to the theory of parallels. His 
Theory of Parallels was written in 1766 and published after his death by 
G. Bernoulli and C. F. Hindenburg ; it is reproduced by Engel and Stackel 
(op. sit. pp. 151 — 208). 

The third part of Lambert's tract is devoted to the discussion of the same 
three hypotheses as Saccheri's, the hypothesis of the right angle being for 
Lambert the first, that of the obtuse angle the second, and that of the acute 
angle the third, hypothesis; and, with reference to a quadrilateral with three 
right angles from which Lambert starts (that is, one of the halves into which 
the median divides Saccheri's quadrilateral), the three hypotheses are the 
assumptions that the fourth angle is a right angle, an obtuse angle, or an 
acute angle respectively. 

Lambert goes much further than Saccheri in the deduction of new 
propositions from the second and third hypotheses. The most remarkable is 
the following. 

The area of a plane triangle, under the second and third hypotheses, is 
proportional to the difference between the sum vf the three angles and two right 
angles. 



h Post. 5] NOTE ON POSTULATE 5 «3 

Thus the numerical expression for the area of a triangle is, under the 
third hypothesis 

&.=;A( r -A-£-C) (1), 

and under the second hypothesis 

&. = *(A + £+C-*) (a), 

where A is a positive constant 

A remarkable observation is appended (5 82) : "In connexion with this it 
seems to be remarkable that the second hypothesis holds if spherical instead of 
plane triangles are taken, because in the former also the sum of the angles is 
greater than two right angles, and the excess is proportional to the area of the 
triangle. 

" It appears still more remarkable that what I here assert of spherical 
triangles can be proved independently of the difficulty of parallels. 

This discovery that the second hypothesis is realised on the surface of a 
sphere is important in view of the development, later, of the Riemann 
hypothesis (1854). 

Still more remarkable is the following prophetic sentence : " I am almost 
inclined to draw the conclusion that the third hypothesis arises with an imaginary 
spherical surface" (cf. Lobachewsky's Gcome'trie imaginaire, 1837). 

No doubt Lambert was confirmed in this by the fact that, in the formula 
(1) above, which, for h = r 1 , represents the area of a spherical triangle, if 
r V- 1 is substituted for r, and r 1 = k, we obtain the formula (1). 

Legend re. 

No account of our present subject would be complete without a full 
reference to what is of permanent value in the investigations of Adrien Marie 
Legendre {1752 — 1833) relating to the theory of parallels, which extended over 
the space of a generation. His different attempts to prove the Euclidean 
hypothesis appeared in the successive editions of his aliments de Giomklrie 
from the first {1794) to the twelfth {1823), which last may be said to contain 
his last word on the subject. Later, in 1833, he published, in the Afhnoires 
de I'Acadimie Royals des Sciences, xn. p. 367 sqq., a collection of his different 
proofs under the title Reflexions sur dffirentes maniires de dhnontrer la thiorie 
des paralteles. His exposition brought out clearly, as Saccheri had done, and 
kept steadily in view, the essential connexion between the theory of parallels 
and the sum of the angles of a triangle. In the first edition of the Elements 
the proposition that the sum of the angles of a triangle is equal to two right 
angles was proved analytically on the basis of the assumption that the choice 
of a unit of length does not affect the correctness of the proposition to be 
proved, which is of course equivalent to Wallis' assumption of the existence of 
similar figures. A similar analytical proof is given in thj notes to the twelfth 
edition. In his second edition Legendre proved Postulate 5 by means of the 
assumption that, given three points not in a straight line, there exists a circle 
passing through all three. In the third edition (1800) he gave the proposition 
that the sum of the angles of a triangle is not greater than two right angles ; 
this proof, which was geometrical, was replaced later by another, the best 
known, depending on a construction like that of Euclid 1. 16, the continued 
application of which enables any number of successive triangles to be evolved 
in which, while the sum of the angles in each remains always equal to the 
sum of the angles of the original triangle, one of the angles increases and the 
sum of the other two diminishes continually. But Legendre found the proof 
of the equally necessary proposition that the sum of the angles of a triangle is 



ai 4 BOOK I [i. Post, S 

not less than two right angles to present great difficulties. He first observed 
that, as in the case of spherical triangles (in which the sum of the angles is 
greater than two right angles) the excess of the sura of the angles over two 
right angles is proportional to the area of the triangle, so in the case of 
rectilineal triangles, if the sum of the angles is less than two right angles by a 
Mrtain deficit, the deficit will be proportional to the area of the triangle. 
Hence if, starting from a given triangle, we could construct another triangle 
in which the original triangle is contained at least m times, the deficit of this 
new triangle will be equal to at least m times that of the original triangle, so 
that the sum of the angles of the greater triangle will diminish progressively 
as m increases, until it becomes zero or negative : which is absurd. The 
whole difficulty was thus reduced to that of the construction of a triangle 
containing the given triangle at least twice ; but the solution of even this 
simple problem requires it to be assumed (or proved) that through a given 
point within a given angle less than two-thirds of a right angle we can always 
draw a straight tine which shall meet both sides of the angle. This is however 
really equivalent to Euclid's Postulate. The proof in the course of which the 
necessity for the assumption appeared is as follows. 

It is required to prove that the sum of the angles of a triangle cannot be 
less than two right angles. 

Suppose A is the least of the three angles of a triangle ABC. Apply to 
the opposite side 2?C a triangle DBC, equal to 
the triangle ACB, and such that the angle 
DBC is equal to the angle ACB, and the angle 
DCB to the angle ABC ; and draw any straight 
line through D cutting AB, AC produced in 
E, F. 

If now the sum of the angles of the triangle 
ABC is less than two right angles, being equal 
to aB-i say, the sum of the angles of the triangle DBC, equal to the 
triangte ABC, is also 2^-8. 

Since the sum of the three angles of the remaining triangles DEB, FDC 
respectively cannot at all events be greater than two right angles [for I>egendre's 
proofs of this see below], the sum of the twelve angles of the four triangles in 
the figure cannot be greater than 

4B + {2B - &) + (2B - ty, i.e. %R-al. 

Now the sum of the three angles at each of the points B, C, D is iR. 

Subtracting these nine angles, we have the result that the three angles of 
the triangle AEF cannot be greater than 2R - 28. 

Hence, if the sum of the angles of the triangle ABC is less than two right 
angles by £, the sum of the angles of the- larger triangle AEF is less than two 
right angles by at least 28. 

We can continue the construction, making a still larger triangle from AEF, 
and so on. 

But, however small 8 is, we can arrive at a multiple 2*$ which shall exceed 
any given angle and therefore tR itself; so that the sum of the three angles 
of a triangle sufficiently large would be zero or even less than zero : which is 
absurd. 

Therefore etc. 

The difficulty caused by the necessity of making the above-mentioned 
assumption made Legendre abandon, in his ninth edition, the method of the 




i. Post. 5] NOTE ON POSTULATE 5 115 

editions from the third to the eighth and return to Euclid's method pure and 
simple. 

But again, in the twelfth, he returned to the plan of constructing any 
number of successive triangles such that the sum of the three angles in all of 
them remains equal to the sum of the three angles of the original triangle, 
but two of the angles of the new triangles become smaller and smaller, while 
the third becomes larger and larger ; and this time he claims to prove in one 
proposition that the sum of the three angles of the original triangle is equal to 
two right angles by continuing the construction of new triangles indefinitely 
and compressing the two smaller angles of the ultimate triangle into nothing, 
while the third angle is made to become a flat angle at the same time. The 
construction and attempted proof are as follows. 

Let ABC be the given triangle ; let A B be the greatest side and BC the 
least ; therefore C is the greatest angle and A the least. 

From A draw AD to the middle point of BC, and produce AD to C, 
making AC equal to AB, 

Produce AB to B 1 , making AB equal to twice AD. 

The triangle ABC is then such that the sum of its three angles is equal 
to the sum of the three angles of the triangle ABC. 




For take AK along AB equal to AD, and join C'K. 

Then the triangles ABD, ACK havt two sides and the included angles 
respectively equal, and are therefore equal in all respects ; and C'K is equal to 
BD or DC. 

Next, in the triangles BCK, A CD, the angles BKC, ADC are equal, 
being respectively supplementary to the equal angles AKC, ADB; and the 
two sides about the equal angles are respectively equal ; 

therefore the triangles BC'K, A CD are equal in all respects. 

Thus the angle AC'B is the sum of two angles respectively equal to the 
angles B, C of the original triangle ; and the angle A in the original triangle 
is the sum of two angles respectively equal to the angles at A and B' in the 
triangle ABC. 

It follows that the sum of the three angles of the new triangle ABC is 
equal to the sum of the angles of the triangle ABC. 

Moreover, the side AC, being equal to AB, and therefore greater than 
AC, is greater than BC which is equal to AC. 

Hence the angle C'AB'w less than the angle ABC ; so that the angle 
CAB is less than \A, where A denotes the angle CAB of the original 
triangle. 

[It will be observed that the triangle ABC is really the same triangle as 
the triangle ABB obtained by the construction of Eucl. 1. 16, but differently 
placed so that the longest side lies along AB.] 

By taking the middle point D of the side BC and repeating the same 
construction, we obtain a triangle AB'C" such that (1) the sum of its three 
angles is equal to the sum of the three angles of ABC, (a) the sum of the 



ai6 BOOK I [1. Post. 5 

two angles CAB", AB"C" is equal to the angle CAB in the preceding 
triangle, and is therefore less than \A, and (3) the angle CAB' is less than 
half the angle CAB, and therefore less than \A. 

Continuing in this way, we shall obtain a triangle Abe such that the sum of 

two angles, those at A and i, is less than — A, and the angle at c is greater 

than the corresponding angle in the preceding triangle. 

If, Legendre argues, the construction be continued indefinitely so that 

- n A becomes smaller than any assigned angle, the point c ultimately ties on 

Alt, and the sum of the three angles of the triangle (which is equal to the sum 
of the three angles of the original triangle) becomes identical with the angle 
at c, which is then a.Jlat angle, and therefore equal to two right angles. 

This proof was however shown to be unsound (in respect of the final 
inference) by J. P. W. Stein in Gergonne's Annaln de Mathimatiques XV., 
1824, pp. 77—79. 

We will now reproduce shortly the substance of the theorems of Legendre 
which are of the most permanent value as not depending on a particular 
hypothesis as regards parallels. 

I. The sum of the three angles of a triangle cannot it greater than two 
right angles. 

This Legendre proved in two ways. 

(r) Mr si proof (in the third edition of the Aliments). 

Let ABC be the given triangle, and ACf a straight line. 

Make CE equal to AC, the angle DCE equal to the angle BAC, and DC 
equal to AB. Join DE. 

Then the triangle DCE is equal to the triangle BAC in all respects. 

If then the sum of the three angles of the triangle ABC is greater than 




%R, the said sum must be greater than the sum of the angles BCA, BCD, 
DCE, which sum is equal to 2H. 

Subtracting the equal angles on both sides, we have the result that 

the angle ABC is greater than the angle BCD. 

But the two sides AB, BC of the triangle ABC are respectively equal to 
the two sides DC, CB of the triangle BCD. 

Therefore the base AC is greater than the base BD (Eucl. 1. 14). 

Next, make the triangle BEG {by the same construction) equal in all 
respects to the triangle BAC or DCE ; and we prove in the same way that 
CE (or AC) is greater than DE. 

And, at the same time, BD is equal to DE, because the angles BCD, 
DEE are equal. 

Continuing the construction of further triangles, however small the 
difference between AC and BD is, we shall ultimately reach some multiple 




i. Post. 5] NOTE ON POSTULATE 5 ai7 

of this difference, represented in the figure by (say) the difference between 
the straight line AJ and the composite line BDFHK, which will be greater 
than any assigned length, and greater therefore than the sum of AB and JJC. 

Hence, on the assumption that the sum of the angles of the triangle ABC 
is greater than 2R, the broken line ABDFHKJ may be less than the straight 
Hne AJ: which is impossible. 

Therefore etc. 

(2) Proof substituted later. 

If possible, let 2^ + a be the sum of the three angles of the triangle ABC, 
of which A is not greater than either of the 
others. 

Bisect BC at H, and produce AH to D, 
making HD equal to AH ; join BD. 

Then the triangles AHC, DHB are equal in 
all respects (l. 4); and the angles CAH,ACHaie 
respectively equal to the angles BDH, DBH. 

It follows that the sum of the angles of the 
triangle ABD is equal to the sum of the angles of the original triangle, i.e. 
to tR + a. 

And one of the angles DAB, ADB is either equal to or less than half the 
angle. CAB. 

Continuing the same construction with the triangle ADB, we find a third 
triangle in which the sum of the angles is still zR + a, while one of them is 
equal to or less than J L CAB)jj\. 

Proceeding in this way, we arrive at a triangle in which the sum of the 
angles is 2R + a, and one of them is not greater than ( L CAB)J2 H . 

And, if n is sufficiently large, this will be less than a. ; in which case we 
should have a triangle in which two angles are together greater than two right 
angles : which is absurd. 

Therefore a is equal to or less than zero. 

(It will be noted that in both these proofs, as in Eucl. 1. 16," it is taken for 
granted that a straight line is infinite in length and does not return into itself, 
which is not true under the Riemann hypothesis.) 

II. On the assumption that the sum of the angles of a triangle is less 
than two right angles, if a triangle is made up of two others, the " deficit" of ' tht, 
former is equal to the sum of the " deficits " of the others. 

In fact, if the sums of the angles of the component triangles are 2R -a, 
2R - fi respectively, the sum of the angles of the whole triangle is 

(zj¥-a) + (2JP-£)-3.ff = 7Je-fa + P). 

III. If the sum of the three angles of a triangle is equal to two right 
angles, the same is true of all triangles obtained by subdividing it by straight 
lines drawn from a vertex to meet the opposite side. 

Since the sum of the angles of the triangle ABC is equal to 2R, if the 
sum of the angles of the triangle ABD were 2R - a, it 
would follow that the sum of the angles of the triangle A 

ADC must be »R + q, which is absurd (by I. above). 

IV. If in a triangle the sum of the three angles is 
equal to two right angles, a quadrilateral can always be 

constructed with four right angles and four equal sides B^ £2 io 

exceeding in length any assigned rectilineal segment. 

Let ABC be a triangle in which the sum of the angles is equal to two 




«i8 BOOK I [i. Post. 5 

right angles. We can assume ABC to be an isosceles right-angled triangle 
because we can reduce the case to this by making subdivisions of ABC by 
straight lines through vertices (as in Prop. III. above). 

Taking two equal triangles of this kind and placing their hypotenuses 
together, we obtain a quadrilateral with four right angles and four equal 
sides. 

Putting four of these quadrilaterals together, we obtain a new quadrilateral 
0/ the same kind but with its sides double of those of the first quadrilateral. 

After n such operations we have a quadrilateral with four right angles and 
four equal sides, each being equal to 3" times the side AH. 

The diagonal of this quadrilateral divides it into two equal isosceles right- 
angled triangles in each of which the sum of the angles is equal to two right 
angles. 

Consequently, from the existence ot one triangle in which the sum of the 
three angles is equal to two right angles it follows that there exists an isosceles 
right-angled triangle with sides greater than any assigned rectilineal segment 
and such that the sum of its three angles is also equal to two right angles. 

V. If the sum of the three angles of one triangle is equal to two right 
angles, the sum of the three angles of any other triangle is also equal to two 
right angles. 

It is enough to prove this for a right-angled triangle, since any triangle can 
be divided into two right-angled triangles. 

Let ABC be any right-angled triangle. 

If then the sum of the angles of any one 
triangle is equal to two right angles, we can 
construct (by the preceding Prop.) an isosceles 
right-angled triangle with the same property and 
with its perpendicular sides, greater than those of 
ABC. 

Let A'BC' be such a triangle, and let it be 
applied to ABC, as in the figure. 

Applying then Prop. 111. above, we deduce 
first that the sum of the three angles of the 
triangle ABC is equal to two right angles, and 
next, for the same reason, that the sum of the three angles of the original 
triangle A BC is equal to two right angles. 

VI. If in any one triangle the sum of the three angles is less than two 
right angles, the sum of the three angles of any other triangle is also less than 
two right angles. 

This follows from the preceding theorem. 

(It will be observed that the last two theorems are included among those 
of Saccheri, which contain however in addition the corresponding theorem 
touching the case where the sum of the angles is greater than two right 
angles.) 

We come now to the bearing of these propositions upon Euclid's Postulate 
5 ; and the next theorem is 

VII. If the sum of the three angles of a triangle is equal to two right 
angles, through any point in a plane there (an only be drawn one parallel to a 
given straight line. 





r. Post. 5] NOTE ON POSTULATE 5 219 

For the proof of this we require the following 

Lemma. // is always possible, through a faint P, to draw a straight line 
which shall make, with a ghten straight line (r), an angle less than any assigned 
angle. 

Let Q be the foot of the perpendicular from /'upon r. 

Let a segment QR be taken on r, 
on either side of Q, such that QR is 
equal to PQ. 

Join PR, and mark off the segment 
RR' equal to PR ; join PR'. 

If tu represents the angle QPR or 
the angle QRP, each of the equal 
angles RPR', RR'P is not greater 
than 10/ 1. 

Continuing the construction, we obtain, after the requisite number of 
operations, a triangle PR,-, R n in which each of the equal angles is equal to 
or less than <o/*". 

Hence we shall arrive at a straight line PR, which, starting from /'and 
meeting r, makes with r an angle as small as we please. 

To return now to the Proposition. Draw from P the straight line s 
perpendicular to PQ. 

Then any straight line drawn from P which meets r in R will form equal 
angles with r and s, since, by hypothesis, the sum of the angles of the triangle 
PQR is equal to two right angles. 

And since, by the Lemma, it is always possible to draw through P straight 
lines which form with r angles as small as we please, it follows that all the 
straight lines through /', except s, will meet r. Hence s is the only parallel 
to r that can be drawn through P. 

The history of the attempts to prove Postulate 5 or something equivalent 
has now been brought down to the parting of the ways. The further 
developments on lines independent of the Postulate, beginning with 
Schweikart {1780 — 1857), Taurinus {1794 — 1874), Gauss {1777 — 1855), 
Lobacbewsky {1793 — 1850), J. Bolyai (180* — 1860), Riemann (1826— 1866), 
belong to the history of non-Euclidean geometry, which is outside the scope 
of this work. I may refer the reader to the full article Sulla teoria delle 
par allele e suite geometric non-euelidec by R. Bonola in Questioni riguardanti 
le matematiche elementari, I., of which I have made considerable use in the 
above, to the same author's La geometria non-euclidea, Bologna, 1906, to the 
first volume of Killing's Einfuhrutig in die Grundlagen der Geometric, 
Paderborn, 1893, to P. Mansion's Premiers princifes de mitag^omitrie, and 
P. Barbarin's La giomitrie ncn-Euclidicnne, Paris, 1902, to the historical 
summary in Veronese's Fondamenti di geometria, 1891, p. 565 sqq., and (for 
original sources) to Engel and Stackel, Die Theorie der Paralleltinicn von 
Eukltd bis auf Gauss, 1895, and Urkunden zur Gcschichte der nickt-Euklidischtn 
Geometric, 1. (Lobachewsky), 1899, and ti. (Wolfgang und Johann Bolyai). 
I will only add that it was Gauss who first expressed a conviction that the 
Postulate could never be proved ; he indicated this in reviews in the Gottin- 
gische gelckrfe Anzeigen, 20 Apr. 18 16 and 18 Oct. 1822, and affirmed it in a 
letter to Bessei of 27 January, 1829. The actual indemonstrability of the Pos- 
tulate was proved by Beltrami (1868) and by Houel {Note sur rimpessibilite" de 
dimontrer par une construction plane leprincipe de la thiorie des paralletes dit Pos- 
tulatum (f.EwA'AmBattaglini's Giornale di matematiehe,\ lit., 1870, pp.84 — 89). 



BOOK I [i. Post. 5 

Alternatives for Postulate 5. 

It may be convenient to collect here a few of the more noteworthy 
substitutes which have from time to time been formally suggested or tacitly 
assumed. 

{ i ) Through a given point only one parallel can be drawn to a given 
straight line or, Two straight lines which intersect one another cannot both be 
parallel to one and the same straight line. 

This is commonly known as " Playfair's Axiom," but it was of course not 
a new discovery. It is distinctly stated in Proclus' note to Eucl. I. 31. 

(r a) If a straight tine intersect one of two parallels, it will intersect the 
other also (Proclus). 

(1 b) Straight lines parallel to the same straight line are parallel to one 
another. 

The forms (1 a) and (1 b) are exactly equivalent to (i). 

{2) There exist straight lines everywhere equidistant from one another 
(Posidonius and Geminus) ; with which may be compared Proclus' tacit 
assumption that Parallels remain, throughout their length, at a finite distance 
from one another. 

(3) There exists a triangle in which the sum of the three angles is equal to 
two right angles (Legendre). 

(4) Given any figure, there exists a figure similar to it of any size toe please 
(Wallis, Carnot, Laplace). 

Saccheri points out that it is not necessary to assume so much, and that it 
is enough to postulate that there exist two unequal triangles with equal angles. 

(5) Through any point within an angle less than two-thirds of a right angle 
a straight tine can always be drawn which meets both sides of the angle 
{Legendre). 

With this may be compared the similar axiom of Lorenz (Grundriss der 
r einen und angewandten Mathematik, 1791): Every straight line through a 
point within an angle must meet one of the sides of the angle. 

(6) Given any three points not in a straight line, there exists a circle passing 
through them {Legendre, W, Bolyai). 

(7) " If I could prove that a rectilineal triangle is possible the content of 
which is greater than any given area, I am in a position to prove perfectly 
rigorously the whole of geometry" (Gauss, in a letter to W. Bolyai, 1 799). 

Cf. the proposition of Legendre numbered iv. above, and the axiom of 
Worpitzky: There exists no triangle in which every angle is as small as we 
please. 

(8) If in a quadrilateral three angles are right angles, the fourth angle is 
a right angle also (Clairaut, 1741). 

(9) If two straight lines are parallel, they arc figures opposite to (or the 
reflex of) one another with respect to the middle points of ail their transversal 
segments (Veronese, Elements, 1904). 

Or, Two parallel straight lines intercept, on every transiiersal which passes 
through the middle point of a segment included between them, another segment 
the middle point of which is the middle point of the first (Ingrami, Etementi, 
1904). 

Veronese and Ingrami deduce immediately Playfair's Axiom, 



NOTES ON THE COMMON NOTIONS 221 



AXIOMS OR COMMON NOTIONS. 

In a paper Sur Pauthenticite des axiomes d'Euclide in the Bulletin des sciences 
math, tt astron. 1884, p. i6z sq. {Mimoires scientifigues, ir., pp, 48—63), Paul 
Tannery maintained that the Common Notions (including the first three) were 
not in Euclid's work but were interpolated later. The following are his main 
arguments. (1) If Euclid had set about distinguishing between indemon- 
strable principles (a) common to all demonstrative sciences and (b) peculiar 
to geometry, he would, says Tannery, certainly not have placed the common 
principles second and the special principles (the Postulates) first. (2) If the 
Common Notions are Euclid's, this designation of them must be his too ; for he 
must have used some name to distinguish them from the Postulates and, if he 
had used another name, such as Axioms, it is impossible to imagine why that 
name was changed afterwards for a less suitable one. The word iwom 
{nation), says Tannery, never signified a notion in the sense of a proposition, 
but a notion of some object ; nor is it found in any technical sense in Plato 
and Aristotle. (3) Tannery's own view was that the formulation of the 
Common Notions dates from the time of Apollonius, and that it was inspired 
by his work relating to the Elements (we know from Proclus that Apollonius 
tried to prove the Common Notions), This idea, Tannery thought, was 
confirmed by a " fortunate coincidence " furnished by the occurrence of the 
word hrovt. {notion) in a quotation by Proclus (p. 100, 6): "we shall agree 
with Apollonius when he says that we have a notion {iwoaui) of a line when 
we order the lengths, only, of roads or walls to be measured." 

In reply to argument (r) that it is an unnatural order to place the purely 
geometrical Postulates first, and the Common Notions, which are not peculiar 
to geometry, last, it may be pointed out that it would surely have been a still 
more awkward arrangement to give the Definitions first and then to separate 
from them, by the interposition of the Common Notions, the Postulates, which 
are so closely connected with the Definitions in that they proceed to postulate 
the existence of certain of the things defined, namely straight lines and circles. 

(2) Though it is true that ii™« in Plato and Aristotle is generally a 
notion of an object, not of a fact or proposition, there are instances in Aristotle 
where it does mean a notion of a fact : thus in the Eth. Nic. ix. 11, 1 1 7 1 a 32 
he speaks of " the notion (or consciousness) that friends sympathise " (1} iwom 
rod m>ya\yth Tok tt>£\uvs) and again, b 14, of "the notion (or consciousness) 
that they are pleased at his good fortune." It is true that Plato and Aristotle 
do not use the word in a technical sense ; but neither was there apparently in 
Aristotle's time any fixed technical term for what we call "axioms," since he 
speaks of them variously as "the so-called axioms in mathematics," "the so- 
called common axioms," " the common (things) " (to. jcotpd), and even " the 
common opinions " (kmvoi bofru). I see therefore no reason why Euclid should 
not himself have given a technical sense to " Common Notions," which is at 
least a distinct improvement upon " common opinions." 

(3) The use of fcroux in Proclus' quotation from Apollonius seems to me 
to be an unfortunate, rather than a fortunate, coincidence from Tannery's point 
of view, for it is there used precisely in the old sense of the notion of an 
object (in that case a line). 

No doubt it is difficult to feel certain that Euclid did himself use the term 
Common Notions, seeing that Proclus' commentary generally speaks of Axioms, 
But even Proclus (p. 194, 8), after explaining the meaning of the word 
"axiom," first as used by the Stoics, and secondly as used by "Aristotle and 



»** BOOK I [!. C. N. i 

the geometers," goes on to say : " For in their view (that of Aristotle and the 
geometers) axiom and common notion are the same thing." This, as it seems 
to me, may be a sort of apology for using the word " axiom " exclusively in 
what has gone before, as if Proclus had suddenly bethought himself that he 
had described both Aristotle and the geometers as using the one term 
" axiom," whereas he should have said that Aristotle spoke of " axioms," while 
"the geometers" (in fact Euclid), though meaning the same thing, called them 
Common Notions, It may be for a like reason that in another passage (p. 76, 
16), after quoting Aristotle's view of an "axiom," as distinct from a postulate 
and a hypothesis, he proceeds : " For it is not by virtue of a common notion 
that, without being taught, we preconceive the circle to be such and such a 
figure." If this view of the two passages just quoted is correct, it would 
strengthen rather than weaken the case for the genuineness of Common Notions 
as the Euclidean term. 

Again, it is clear from Aristotle's allusions to the " common axioms in 
mathematics " that more than one axiom of this kind had a place in the text- 
books of his day ; and as he constantly quotes the particular axiom that, if 
equals be taken from equals, the remainders are equals which is Euclid's Common 
Notion 3, it would seem that at least the first three Common Notions were 
adopted by Euclid from earlier textbooks. It is ? besides, scarcely credible 
that, if the Common Notions which Apollonius tried to prove had not been 
introduced earlier (e.g. by Euclid), they would then have been interpolated as 
axioms and not as propositions to be proved. The line taken by Apollonius 
is much better explained on the assumption that he was directly attacking 
axioms which he found already admitted into the Elements. 

Proclus, who recognised the five Common Notions given in the text, warns 
us, not only against the error of unnecessarily multiplying the axioms, but 
against the contrary error of reducing their number unduly (p. 196, 15), "as 
Heron does in enunciating three only; for it is also an axiom that the whole is 
greater than the part, and indeed the geometer employs this in many places for 
his demonstrations, and again that things which coincide are equal." 

Thus Heron recognised the first three of the Common Notions ; and this 
fact, together with Aristotle's allusions to "common axioms" (in the plural), 
and in particular to our Common Notion 3, may satisfy us that at least the first 
three Common Notions were contained in the Elements as they left Euclid's 
hands. 

Common Notion i. 

Things which are equal to the same thing are also equal to one another. 

Aristotle throughout emphasises the fact that axioms are self-evident truths, 
which it is impossible to demonstrate. If, he says, any one should attempt to 
prove them, it could only be through ignorance. Aristotle therefore would 
undoubtedly have agreed in Proclus' strictures on Apollonius for attempting 
to prove the axioms. Proclus gives (p. 194, 25), as a specimen 
of these attempted proofs by Apollonius, that of the first of the 
Common Notions. " Let A be equal to B, and the latter to C; 
I say that A is also equal to C. For, since A is equal to JB, it A B 
occupies the same space with it ; and since B is equal to C, it 
occupies the same space with it. 

Therefore A also occupies the same space with C." 

Proclus rightly remarks (p. 194, 23) that "the middle term is no more 



I. C.N 1—3] NOTES ON COMMON NOTIONS 1—3 *»3 

intelligible {better known, yvmpifuar€pov} than the conclusion, if it is not 
actually more disputable." Again (p. 195, 6), the proof assumes two things, 
(1) that things which "occupy the same space" <tot<k) are equal to one 
another, and (2) that things which occupy the same space with one and the 
same thing occupy the same space with one another ; which is to explain the 
obvious by something much more obscure, for space is an entity more 
unknown to us than the things which exist in space. 

Aristotle would also have objected to the proof that it is partial and not 
general {xatfoXav), since it refers only to things which can be supposed to 
occupy a space (or take up room), whereas the axiom is, as Froclus says 
(p. 196, 1), true of numbers, speeds, and periods of time as well, though of 
course each science uses axioms in relation to its own subject-matter only. 

Common Notions 2, 3. 

2, Kttt lav iiTiia «ra nrpo<rr*9j7, ™ oka JVfw urn. 

3. Kai iav otto four lira dtfxHpffljj, ta narnAtHTO/itni fUTtk Ifftt. 

7, JftquaU be added to equals, the wholes are equal. 

3. //equals he subtracted from equals, the remainders are equal. 

These two Common Notions are recognised by Heron and Proclus as 
genuine. The latter is the axiom which is so favourite an illustration, with 
Aristotle. 

Following them in the Mss. and editions there came four others of the 
same type as 1 — 3. Three of these are given by Heiberg in brackets; the 
fourth he omits altogether. 

The three are : 

(«) If equals be ad/led to unequals, the wholes are unequal, 

(b) Things which are double a/ the same thing are equal to oHe another. 

{c) Things which are halves of the same thing are equal to one another. 

The fourth, which was placed between (a) and {&), was : 

(d) If equals be subtracted front unequals, the remainders are unequal. 

Proclus, in observing that axioms ought not to be multiplied, indicates 
that all should be rejected which follow from the five admitted by him and 
appearing in the text above {p. 155). He mentions the second of those just 
quoted (/') as one of those to be excluded, since it follows from Common 
Notion 1. Proclus does not mention (a), (c) or (rf); an-Naiml gives (a), (d), {b) 
and (f), in that order, as Euclid's, adding a note of Simplicius that " three 
axioms (sentenriae acceptae) only are extant in the ancient manuscripts, but 
the number was increased in the more recent." 

(a) stands self-condemned because " unequal " tells us nothing. It is easy 
to see what is wanted if we refer to 1. 17, where the same angle is added to a 
greater and a less, and it is inferred that the first sum is greater than the second. 
So far however as the wording of (a) is concerned, the addition of equal to 
greater and less might be supposed to produce less and greater respectively. If 
therefore such an axiom were given at all, it should be divided into two. 
Heiberg conjectures that this axiom may have been taken from the commentary 
of Pappus, who had the axiom about equals added to unequais quoted below 
(e); if so, it can only be an unskilful adaptation of some remark of Pappus, for 
his axiom (e) has some point, whereas (a) is useless. 

As regards (b), I agree with Tannery in seeing no sufficient reason why, if 



«4 BOOK I [i. C. N. a— 4 

we reject it (as we certainly must), the words in I. 47 " But things which are 
double of equals are equal to one another " should be condemned as an 
interpolation. If they were interpolated, we should have expected to find the 
same interpolation in ). 42, where the axiom is tacitly assumed. I think 
it quite possible that Euclid may have inserted such words in one case and 
left them out in another, without necessarily implying either that he was 
quoting a formal Common Notion of his own or that he had wt included 
among his Common Notions the particular fact stated as obvious. 

The corresponding axiom (c) about the halves of equals can hardly be 
genuine if {b) is not, and Proclus does not mention it. Tannery acutely 
observes however that, when Heiberg, in 1. 37, 38, brackets words stating that 
"the halves of equal things are equal to one another" on the ground that 
axiom (e) was interpolated (although before Theon's time), and explains that 
Euclid used Common Notion 3 in making his inference, he is clearly mistaken. 
For, while axiom (b) is an obvious inference from Common Notion 2, axiom (c) 
is not an inference from Common Notion 3. Tannery says, in a note, that (c) 
would have to be established by rtdudio ad absurdum with the help of axiom 
(6), that is to say, of Common Notion 2. But, as the hypothesis in the reductw 
ad absurdum would be that one of the halves is greater than the other, and it 
would therefore be necessary to prove that the one whole is greater than the 
other, while axiom (b) or Common Notion 2 only refers to equals, a little 
argument would be necessary in addition to the reference to Common Notion 2. 
I think Euclid would not have gone through this process in order to prove (c), 
but would have assumed it as equally obvious with (b). 

Proclus (pp. 197, 6 — 198, 5) definitely rejects two other axioms of the 
above kind given by Pappus, observing that, as they follow from the genuine 
axioms, they are rightly omitted in most copies, although Pappus said that 
they were " on record " with the others (<rvvara.ypiufn<T$at) : 

(e) If unequa/s be added to equals, the difference between the wholes is equal 
to the difference between the added parts ; and 

(/) //equals be added to unequals, the difference between the wholes is equal 
to the difference bttween (he original unequals. 

Proclus and Simplicius (in an-Nairizi) give proofs of both. The proof of 
the former, as given by Simplicius, is as follows : 

Let AB, CD be equal magnitudes ; and let EB, FD be £ 
added to them respectively, EB being greater than FD. q 

I say that AE exceeds CF by the same difference as that by 
which BE exceeds DF. 

Cut off from BE the magnitude BG equal to DF. 

Then, since AE exceeds AG by GE, and AG is equal to CF 
and BG to DF, 

AE exceeds CF by the same difference as that by which BE 
exceeds DF. 

Common Notion 4. 

Kcu to. t<f>upyjj(,uvra tir oAAipXa ura aAAr/Xoi? cotcV. 

Things which coincide with one another are equal to one anotJur. 

The word itjmp/Aiiltiv, as a geometrical term, has a different meaning 
according as it is used in the active or in the passive. In the passive, 
<£af>/u>£«rltu, it means "to be applied to" without any implication that the 
applied figure will exactly fit, or coincide with, the figure to which it is applied ; 
on the other hand the active 'v^ap^t^w is used intransitively and means " to 



i. C. N. 4] NOTES ON COMMON NOTIONS 2—4 225 

lit exactly," " to coincide with." In Euclid and Archimedes t<#>a/)^o{<«' is 
constructed with «ri and the accusative, in Pappus with the dative. 

On Common Notion 4 Tannery observes that it is incontestably geometrical 
in character, and should therefore have been excluded from the Common 
Notions; again, it is difficult to see why it is not accompanied by its converse, 
at all events for straight lines (and, it might be added, angles also), which 
Euclid makes use of in I. 4. As it is, says Tannery, we have here a definition 
of geometrical equality more or less sufficient, but not a real axiom. 

It is true that Froclus seems to recognise this Common Notion and the next 
as proper axioms in the passage (p. 196, 15 — 21) where he says that we should 
not cut down the axioms to the minimum, as Heron does in giving only three 
axioms; but the statement seems to rest, not upon authority, but upon an 
assumption that Euclid would state explicitly at the beginning all axioms 
subsequently used and not reducible to others unquestionably included. Now 
in 1. 4 this Common Notion is not quoted ; it is simply inferred that " the base 
BC will coincide with EF, and will be equal to it." The position is therefore 
the same as it is in regard to the statement in the same proposition that, "if... 
the base BC does not coincide with EF, two straight lines will enclose a spate : 
which is impossible " ; and, if we do not admit that Euclid had the axiom that 
" two straight lines cannot enclose a space," neither need we infer that he had 
Common Notion 4. I am therefore inclined to think that the latter is more 
likely than not to be an interpolation. 

It seems clear that the Common Notion, as here formulated, is intended 
to assert that superposition is a legitimate way of proving the equality of two 
figures which have the necessary parts respectively equal, or, in other words, 
to serve as an axiom of congruence. 

The phraseology of the propositions, e.g. 1. 4 and 1. 3, in which Euclid 
employs the method indicated, leaves no room for doubt that he regarded one 
figure as actually moved and placed upon the other. Thus in 1. 4 he says, 
" The triangle ABC being applied (t<£apfu>{o(ia>ov) to the triangle DEF, and 
the point A being placed {rJltjitvov) upon the point D, and the straight line 
AB on DE, the point B will also coincide with E because AB is equal to 
DE"; and in 1. 8, "If the sides BA, AC do not coincide with ED, DF, but 
fall beside them (take a different position, TapaAAafowrtv), then " etc. At the 
same time, it is clear that Euclid disliked the method and avoided it wherever 
he could, e.g. in 1. 26, where he proves the equality of two triangles which have 
two angles respectively equal to two angles and one side of the one equal to 
the corresponding side of the other. It looks as though he found the method 
handed down by tradition (we can hardly suppose that, if Thales proved that 
the diameter of a circle divides it into two equal parts, he would do so by any 
other method than that of superposition), and followed it, in -the few cases 
where he does so, only because he had not been able to see his way to a 
satisfactory substitute. But seeing how much of the Elements depends on 1. 4, 
directly or indirectly, the method can hardly be regarded as being, in Euclid, 
of only subordinate importance , on the contrary, it is fundamental. Nor, as 
a matter of fact, do we find in the ancient geometers any expression of doubt 
as to the legitimacy of the method. Archimedes uses it to prove that any 
spheroidal figure cut by a plane through the centre is divided into two equal 
parts in respect of both its surface and its volume; he also postulates in 
Equilibrium of Planes 1. that "when equal and similar plane figures coincide 
if applied to one another, their centres of gravity coincide also." 

Killing {Einfuhrung in die Grundlagen der Geometric, 11. pp. 4, 5) 



2 26 BOOK I [i. C. M 4 

contrasts the attitude of the Greek geometers with that of the philosophers, 
who, he says, appear to have agreed in banishing motion from geometry 
altogether. In support of this he refers to the view frequently expressed by 
Aristotle that mathematics has to do with immovable objects {wivr/rd), and that 
only where astronomy is admitted as part of mathematical science is motion 
mentioned as a subject for mathematics. Cf. Mttaph. 989 b 32 "For mathe- 
matical objects are among things which exist apart from motion, except such 
as relate to astronomy"; Metapk. rofi4 a 30 "Physics deals with things 
which have in themselves the principle of motion ; mathematics is a theoretical 
science and one concerned with things which are stationary (^mfovto) but not 
separable" (sc. from matter); in Pkysies if. z, 193 b 34 he speaks of the 
subjects of mathematics as "in thought separable from motion." 

But I doubt whether in Aristotle's use of the words " immovable," " with- 
out motion " etc. as applied to the subjects of mathematics there is any 
implication such as Killing supposes. We arrive at mathematical concepts 
by abstraction from material objects; and just as we, in thought, eliminate 
the matter, so according to Aristotle we eliminate the attributes of matter as 
such, e.g. qualitative change and motion. It does not appear to me that the 
use of " immovable " in the passages referred to means more than this. I do 
not think that Aristotle would have regarded it as illegitimate to move a 
geometrical figure from one position to another ; and I infer this from a 
passage in De caelo lit. 1 where he is criticising "those who make up every 
body that has an origin by putting together plants, and resolve it again into 
planes." The reference must be to the Timaeus (54 b sqq.) where Plato 
evolves the four elements in this way. He begins with a right-angled triangle 
in which the hypotenuse is double of the smaller side; six of these put together 
in the proper way produce one equilateral triangle. Making solid angles with 
{a) three, (6) four, and (c) five of these equilateral triangles respectively, and 
taking the requisite number of these solid angles, namely four of (a), six of (&) 
and twelve of (c) respectively, and putting them together so as to form regular 
solids, he obtains (a) a tetrahedron, (/J) an octahedron, (v) an icosahedron 
respectively. For the fourth element (earth), four isosceles right-angled triangles 
are first put together so as to form a square, and then six of these squares are 
put together to form a cube. Now, says Aristotle (299 b 23), "it is absurd that 
planes should only admit of being put together so as to touch in a line; for just 
as a line and a line are put together in both ways, lengthwise and breadthwise, 
so must a plane and a plane. A line can be combined with a line in the sense 
of being a line superposed, and not added"; the inference being that a plane can 
be superposed on &plane. Now this is precisely the sort of motion in question 
here; and Aristotle, so far from denying its permissibility, seems to blame 
Plato for not using it. Cf. also Physits v. 4, 228 b 25, where Aristotle speaks 
of "the spiral or other magnitude in which any part will not coincide with 
any other part," an where superposition is obviously contemplated. 

Motion without deformation. 

It is well known that Helmholtz maintained that geometry requires us to 
assume the actual existence of rigid bodies and their free mobility in space, 
whence he inferred that geometry is dependent on mechanics. 

Veronese exposed the fallacy in this {Fondamenti ii geometria, pp. xxxv — 
xxxvi, 239 — 240 note, 615 — 7), his argument being as follows. Since geometry 
is concerned with empty space, which is immovable, it would be at least strange 
if it was necessary to have recourse to the real motion of bodies for a definition, 



]. C. N. 4] NOTE ON COMMON NOTION 4 xa-j 

and for the proof of the properties, of immovable space. We must distinguish 
the intuitive principle of motion in itself from that of motion without deforma- 
tion. Every point of a figure which moves is transferred to another point in 
space. " Without deformation " means that the mutual relations between the 
points of the figure do not change, but the relations between them and other 
figures do change (for if they did not, the figure could not move). Now 
consider what we mean by saying that, when the figure A has moved from 
the position A, to the position A it the relations between the points of A in 
the position A, are unaltered from what they were in the position A u are the 
same in fact as if A had not moved but remained at A t . We can only say 
that, judging of the figure (or the body with its physical qualities eliminated) 
by the impressions it produces in us during its movement, the impressions 
produced in us in the two different positions (which are in time distinct) 
are equal. In fact, we are making use of the notion of equality between two 
distinct figures. Thus, if we say that two bodies are equal when they 
can be superposed by means of movement without deformation, we are com- 
mitting a petitio principii. The notion of the equality of spaces is really prior 
to that of rigid bodies or of motion without deformation. HelmholU supported 
his view by reference to the process of measurement in which the measure 
must be, at least approximately, a rigid body, but the existence of a rigid body 
as a standard to measure by, and the question how we discover two equal 
spaces to be equal, are matters of no concern to the geometer. The method 
of superposition, depending on motion without deformation, is only of use as 
a practical test ; it has nothing to do with the theory of geometry. 

Compare an acute observation of Schopenhauer {Die Welt als (Vi/le, 2 ed- 
1844, 11. p. 130) which was a criticism in advance of Helmholtz' theory : "I 
am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the 
eighth is not rather attacked : ' Figures which coincide (sich decken) are 
equal to one another.' For coincidence (das Sichdecken) is either mere 
tautology, or something entirely empirical, which belongs, not to pure intuition 
(Anschauung), but to external sensuous experience. It presupposes in fact 
the mobility of figures ; but that which is movable in space is matter and 
nothing else. Thus this appeal to coincidence means leaving pure space, the 
sole element of geometry, in order to pass over to the material and empirical." 

Mr Bertrand Russell observes {Encyclopaedia Britannica, Suppl. Vol. 4, 
1Q02, Art. " Geometry, non-Euclidean ") that the apparent use of motion here 
is deceptive ; what in geometry is called a motion is merely the transference 
of our attention from one figure to another. Actual superposition, which is 
nominally employed by Euclid, is not required; all that is required is the 
transference of our attention from the original figure to a new one defined by 
the position of some of its elements and by certain properties which it shares 
with the original figure. 

If the method of superposition is given up as a means of defining theoreti- 
cally the equality of two figures, some other definition of equality is necessary. 
But such a definition can be evolved out of empirical or practical observation 
of the result of superposing two material representations of figures. This is 
done by Veronese {Elementi di geometria, 1904) and Ingrami {Element i di 
geometria, 1904). Ingrami says, namely (p. 66); 

" If a sheet of paper be folded double, and a triangle be drawn Upon it 
and then cut out, we obtain two triangles superposed which we in practice call 
equal. If points A, B, C, D ... be marked on one of the triangles, then, 
when we place this triangle upon the other (so as to coincide with it), we see 



1*8 BOOK I [i. C. If. 4 

chat each of the particular points taken on the first is superposed on one 
particular point of the second in such a way that the segments AB, AC, AD, 
BC, BD, CD, ... ace respectively superposed on as many segments in the 
second triangle and are therefore equal to them respectively. In this way we 
justify the following 

"Definition of equality. 

" Any two figures whatever will be called equal when to the points of one 
the points of the other can be made to correspond univocally [i.e. every one 
point in one to one distinct point in the other and vice versa] in such a way 
that the segments which join the points, two and two, in one figure are 
respectively equal to the segments which join, two and two, the corresponding 
points in the other." 

Ingram! has of course previously postulated as known the signification of 
the phrase equal (reciiUneat) segments, of which we get a practical notion when 
we can place one upon the other or can place a third movable segment 
successively on both. 

New systems of Congruence-Postulates. 

In the fourth Article of Questioni riguardanti ie matematiehe etementari, I., 
pp. 93 — 122, a review is given of three different systems : (i) that of Pasch in 
Vorlesungen titer neuere Geometrie, 1882, p. 101 sqq., (3) that of Veronese 
according to the Fondamenti di geometria, 1891, and the Ekmcnti taken 
together, (3) that of Hilbert (see Grundlagen der Geometric, 1903, pp. 7—15). 

These systems differ in the particular conceptions taken by the three 
authors as primary, (t) Pasch considers as primary the notion of congruence 
or equality between any figures which are made up of a finite number 0/ points 
only. The definitions of congruent segments and of congruent angles have to 
be deduced in the way shown on pp. 102 — 103 of the Article referred to, after 
which Eucl. 1. 4 follows immediately, and Eucl. 1. 26 (1) and 1. 8 by a method 
recalling that in Eucl. 1. 7, 8. 

(2) Veronese takes as primary the conception of congruence between 
segments (rectilineal). The transition to congruent angles, and thence to 
triangles is made by means of the following postulate: 

"Let AB, ^Cand AB, A'C be two pairs of straight lines intersecting 
at A, A', and let there be determined upon them the congruent segments 
AB, A'ff and the congruent segments AC, A'C ; 

then, if BC, BC are congTuent, the two pairs of straight lines are con- 
gruent." 

<i) Hilbert takes as primary the notions of congruence between both 
segments and angles. 

It is observed in the Article referred to that, from the theoretical stand- 
point, Veronese's system is an advance upon that of Pasch, since the idea of 
congruence between segments is more simple than that of congruence between 
any figures; but, didactically, the development of the theory is more compli- 
cated when we start from Veronese's system than when we start from that of 
Pasch. 

The system of Hilbert offers advantages over both the others from the 
point of view of the teaching of geometry, and I shall therefore give a short 
account of his system only, following the Artiole above quoted. 



i. C. JV. 4] NOTE ON COMMON NOTION 4 339 






Hilbcrt's system. 

The following are substantially the Postulates laid down, 

(1) If one segment is congruent toith another, the second is also congruent 
with the first. 

(2) If an angle is congruent with another angle, the second angle is also 
congruent with the first. 

(3) Two segments congruent with a third are congruent with one another. 

(4) Two angles congruent with a third are congruent with one another. 

(5) Any segment AB is congruent with itself, independently of its sense. 
This we may express symbolically thus : 

AB=AB= BA. 

(6) Any angle (ab) is congruent with itself, independently of its sense. 
This we may express symbolically thus : 

(ab) = (ab) = (6a). 

(7) On any straight line x\ starting from any one of its points A', and on 
each side of it respectively, there exists one and only one segment congruent with a 
segment AB belonging to the straight tine 1. 

(&) Given a ray a, issuing from a paint O, in any plane which contains it 
and on each of the two sides of it, there exists one and only one ray b issuing 
from O such that the angle (ab) is congruent with a given angle (a'b'). 

(9) If AB, BC are two consecutive segments of the same straight line r 
(segments, that is, having an extremity and no other point common), and A'B', 
B'C two consecutive segments on another straight line r", and if AB E A'B', 
BC 3 B'C, then 

AC=A'C. 

{10) If (ah), (be) are two consecutive angles in the same planer (angles, 
that is, having the vertex and one side common), and (a'b'), (b'c') two consecu- 
tive angles in another plane ■*■', and if (ab) £ (a'b'), (be) = (b'c'), then 

(ac) = (a'c'). 

(n) If two triangles have two sides and the included angles respectively 
congruent, they have also their third sides congruent as well as the angles 
opposite to the congruent sides respectively. 

As a matter of fact, Hilbert's postulate corresponding to (n) does not 
assert the equality of the third sides in each, but only the equality of the two 
remaining angles in one triangle to the two remaining angles in the other 
respectively. He proves the equality of the third sides (thereby completing 
the theorem of Eucl. L 4) by reductio 

ad absurdum thus. Let ABC, A'B'C A A' 

be the two triangles which have the S\ jr\ 

sides AB, AC respectively congruent ^r \ jr jl 

with the sides A B', A'C and the j^ \ y^ ; \ 

included angle at A congruent with ^— £ gf- £-jy 

the included angle at A'. 

Then, by Hilbert's own postulate, the angles ABC, A'B'C are congruent, 
as also the angles ACB, A'C'ff. 

If BC is not congruent with B'C, let D be taken on BC such that BC, 
BD are congruent and join A'D. 



330 BOOK I [i. C. N. 4 

Then the two triangles ABC, A'ffD have two sides and the included 
angles congruent respectively ; therefore, by the same postulate, the angles 
BAC, SA'D are congruent 

But the angles BAC, B'A'C' are congruent; therefore, by (4) above, the 
angles B'A'C 1 , BAD are congruent : which is impossible, since it contradicts 
(8) above. 

Hence BC, B'C cannot but be congruent. 

Eucl. 1. 4 is thus proved ; but it seems to be as well to include all of that 
theorem in the postulate, as is done in (11) above, since the two parts of it are 
equally suggested by empirical observation of the result of one superposition. 

A proof similar to that just given immediately establishes Eucl. 1. 26 (1), 
and Hilbert next proves that 

If two angles ABC, A' B'C' are congruent with one another, their supple- 
mentary angles CBD, C'B'D' are also congruent with one another. 

We choose A, D on one of the straight lines forming the first angle, and 
A', & on one of those forming the second angle, and again C, C on the other 




* Vi 0' 

straight lines forming the angles, so that A'B is congruent with AB, C'ff 
with CB, and Uff with DB. 

The triangles ABC, A' B'C are congruent, by (11) above; and AC is 
congruent with A'C, and the angle CAB with the angle C'A'B 1 . 

Thus, AD, A'D' being congruent, by (9), the triangles CAD, C'A'D are 
also congruent, by ( 1 1 ) ; 

whence CD is congruent with CD?, and the angle ADC with the angle 
ADC. 

Lastly, hy (n), the triangles CDS, CDS are congruent, and the angles 
CBD, C'B'D are thus congruent. 

Hilbert's next proposition is that 

Given that the angle (h, k) in the plane a is congruent ■with the angle (h', k') 
in the plane a', and that 1 is a half-ray in the plane a. starting from the vertex 
of the angle (h, k) and lying within that angle, there always exists a half-ray V 
in the second plane a', starting from the vertex of the angle {h', k') and lying 
within that angle, such that 

(h,l)s(h',l'), and(k, l) = (k',I'). 

If O, C are the vertices, we choose points A, B on h, k, and points A, ff 
on H, k' respectively, such that OA, OA are congruent and also OB, Off. 




The triangles OAB, OAff are then congruent ; and, if / meets AB in C, 
: can determine C on A'B such that AC is congruent with AC. 
Then f drawn from through C is the half-ray required. 



I. C. N. 4] 



NOTE ON COMMON NOTION 4 



S31 



The congruence of the angles (h, I), (A', /) follows from (11) directly, and 
that of {k, I) and {X, t) follows in the same way after we have inferred by 
means of (9) that, AB, AC being respectively congruent with A'B 1 , A'C, the 
difference BC is congruent with the difference B C'. 

It is by means of the two propositions just given that Hilbert proves that 
All right angles are congruent with one another. 

Let the angle BAD be congruent with its adjacent angle CAD, and 
likewise the angle BA'D congruent with its adjacent angle CA'D All four 
angles are then right angles. 







A' 



C 



If the angle B'A'D is not congruent with the angle BAD, let the angle 
with AB for one side and congruent with the angle B'A'D be the angle 
BAD', so that AD' falls either within the angle BAD or within the angle 
DAC. Suppose the former. 

By the last proposition but one (about adjacent angles), the angles 
BAD, BAD' being congruent, the angles CA'D, CAD' are congruent. 

Hence, by the hypothesis and postulate (4) above, the angles BAD' , 
CAD" are alio congruent. 

And, since the angles BAD, CAD are congruent, we can find within the 
ingle CAD a half-ray CAD"' such that the angles BAD', CAD" are 
congruent, and likewise the angles DAD', DAD" (by the last proposition). 

But the angles BAD', CAD" were congruent (see above); and it 
follows, by (4), that the angles CAD', CAD" are congruent : which is 
impossible, since it contradicts postulate (8), 

Therefore etc. 

Euclid 1. s follows directly by applying the postulate (n) above to ABC, 
ACB as distinct triangles. 

Postulates (9), (10) above give in substance the proposition that "the 
sums or differences of segments, or of 
angles, respectively equal, are equal." 

Lastly, Hilbert proves Eucl. 1. 8 by 
means of the theorem of Eucl. 1. 5 and 
the proposition just stated as applied to 
angles. 

ABC, A'BC being the given triangles 
with three sides respectively congruent, 
we suppose an angle CBA" to be deter- 
mined, on the side of BC opposite to A, 
congruent with the angle A'BC, and we make BA" equal to A'B. 

The proof is obvious, being equivalent to the alternative proof often given 
in our text-books for Eucl. I. 8. 










a 3 « BOOK I [[. C. N. 5 

Common Notion 5. 

koI to oAov tou ttipavs fLttfcav [f tTTtv], 
J>fe icAoA w greater than the part. 

Prod us includes this " axiom " on the same ground as the preceding one. 
I think however there is force in the objection which Tannery takes to it, 
namely that it replaces a different expression in Eucl. f. 6, where it is stated 
that " the triangle DBC will be equal to the triangle A CB, the /ess to the 
greater: which is absurd." The axiom appears to be an abstraction or 
generalisation substituted for an immediate inference from a geometrical 
figure, but it takes the form of a sort of definition of whole and part. The 
probabilities seem to be against its being genuine, notwithstanding Proclus' 
approval of it 

Clavius added the axiom that the whole is the equal to the sum of its parts. 

Other Axioms introduced after Euclid's time. 

[9] Two straight lines do not enclose (or can tain) a space. 
Proclus {p. 196, 21) mentions this in illustration of the undue multiplication 
of axioms, and he points out, as an objection to it, that it belongs to the 
subject matter of geometry, whereas axioms are of a general character, and 
not peculiar to any one science. The real objection to the axiom is that it is 
unnecessary, since i.he fact which it states is included in the meaning of 
Postulate i. It was nr> doubt taken from the passage in 1. 4, "if. the base 
BC does not coincide with the base EF, two straight tines wilt enclose a space : 
which is impossible"; and we must certainly regard it as an interpolation, 
notwithstanding that two of the best mss. have it after Postulate 5, and one 
gives it as Common Notion 9. 

Pappus added some others which Proclus objects to (p. 198, 5) because 
they are either anticipated in the definitions or follow from them. 

(g) All the parts of a plane, or of a straight line, coincide with one another. 
\K) A point divides a line, a line a surface, and a surface a solid; on which 
Proclus remarks that everything is divided by the same things as those by 
which it is bounded. 

An-Nairizi {ed. Besthorn-Heiberg, p. 31, ed. Curtze, p. 38) in his version 
of this axiom, which be also attributes to Pappus, omits the reference to 
solids, but mentions planes as a particular case of surfaces. 
" (a) A Surface cuts a surface in a line ; 
fj8) If two surfaces which cut one another are plane, they cut one another 

in a straight line ; 

(y) A line cuts a line in a point (this last we need in the first proposition}." 

(A) Magnitudes are susceptible of the infinite {or unlimited) both by way of 

addition and by way of successive diminution, but in both cases potentially only 

(to mrttpov fv rots fnyi$ttrw iirriv got Tp xpoaGitrtt Kal rp iiittai0atpio , u, oW ifLtt 

Si imLrtpav). 

An-Nairizi's version of this refers to straight lines and plane surfaces only : 
"as regards the straight line and the plane surface, in consequence of their 
evenness, it is possible to produce them indefinitely. 

This "axiom" of Pappus, as quoted by Proclus, seems to be taken directly 
from the discussion of to avtipov in Aristotle, Physics ill. 5 — 8, even to the 
wording, for, while Aristotle uses the term division {huaiptatsi) most frequently 
as the antithesis of addition (avrBttrn), he occasionally speaks of subtraction 
(AAJHupttrn) and diminution (icaflaipHrii). Hankel (Zur Geschichte der Mathe- 
matik im Alterthum und Mittelalter, 1874, pp. 119 — 120) gave an admirable 



1. Axx.] ADDITIONAL AXIOMS 233 

summary of Aristotle's views on this subject ; and they are stated in greater 
detail in Gorland, AristoteUs und die Mathematik, Marburg, 1899, pp. 157— 
183. The infinite or unlimited (airttpov) only exists potentially {Sura/i«), not 
in actuality (Ivtpytiq.). The infinite is so in virtue of its endlessly changing 
into something else, like day or the Olympic Games {Phys. m. 6, 206 a 15 — 2 S)- 
The infinite is manifested in different forms in time, in Man, and in the 
division of magnitudes. For, in general, the infinite consists in something new 
being continually taken, that something being itself always finite but always 
different. Therefore the infinite must not be regarded as a particular thing 
(toS« t»), as man, house, but as being always in course of becoming or decay, 
and, though finite at any moment, always different from moment to moment. 
But there is the distinction between the forms above referred to that, whereas 
in the case of magnitudes what is once taken remains, in the case of time and 
Man it passes or is destroyed but the succession is unbroken. The case of 
addition is in a sense the same as that of division ; in the finite magnitude the 
former takes place in the converse way to the latter ; for, as we see the finite 
magnitude divided ad infinitum, so we shall find that addition gives a sum 
tending to a definite limit. I mean that, in the case of a finite magnitude, 
you may take a definite fraction of it and add to it (continually) in the same 
ratio ; if now the successive added terms do not include one and the same 
magnitude whatever it is [i.e. if the successive terms diminish in geometrical 
progression], you will not come to the end of the finite magnitude, but, if the 
ratio is increased so that each term does include one and the same magnitude 
whatever it is, you will come to the end of the finite magnitude, for every 
finite magnitude is exhausted by continually taking from it any definite 
fraction whatever. Thus in no other sense does the infinite exist, but only 
in the sense just mentioned, that is, potentially and by way of diminution 
(106 a 25 — b 13). And in this sense you may have potentially infinite 
addition, the process being, as we say, in a manner, the same as with division 
ad infinitum : for in the case of addition you will always be able to find some* 
thing outside the total for the time being, but the total will never exceed every 
definite (or assigned) magnitude in the way that, in the direction of division, 
the result will pass every definite magnitude, that is, by becoming smaller 
than it. The infinite therefore cannot exist even potentially in the sense of 
exceeding every finite magnitude as the result of successive addition (206 b 
16 — 22). It follows that the correct view of the infinite is the opposite of 
that commonly held : it is not that which has nothing outside it, but that 
which always has something outside it (206 b 33 — 207 a r). 

Contrasting the case of number and magnitude, Aristotle points out that 
(1) in number there is a limit in the direction of smallness, namely unity, but 
none in the other direction : a number may exceed any assigned number 
however great ; but (2) with magnitude the contrary is the case : you can find 
a magnitude smaller than any assigned magnitude, but in the other direction 
there is no such thing as an infinite magnitude (207 b r — 5). The tatter 
assertion he justified by the following argument. However large a thing can 
be potentially, it can be as large actually. But there is no magnitude 
perceptible to sense that is infinite. Therefore excess over every assigned 
magnitude is an impossibility j otherwise there would be something larger 
than the universe (oipavot) {207 b 17—21). 

Aristotle is aware that it is essentially of physical magnitudes that he is 
speaking. He had observed in an earlier passage (PAys, in, 5, 204 a 34) that 
it ts perhaps a more general inquiry that would be necessary to determine 



*34 BOOK ! [i. Axx. 

whether the infinite is possible in mathematics, and in the domain of thought 
and of things which have no magnitude ; but he excuses himself from entering 
upon this inquiry on the ground that his subject is physics and sensible 
objects. He returns however to the bearing of his conclusions on mathematics 
in m. 7, 207 b 2j : "my argument does not even rob mathematicians of their 
study, although it denies the existence of the infinite in the sense of actual 
existence as something increased to such an extent that it cannot be gone 
through (dSwfiTifrov) ; for, as it is, they do not even need the infinite or use 
it, but only require that the finite (straight line) shall be as long as they please; 
and another magnitude of any size whatever can be cut in the same ratio as 
the greatest magnitude. Hence it will make no difference to them for the 
purpose of demonstration." 

I^astly, if it should be urged that the infinite exists in thought, Aristotle 
replies that this does not involve its existence in fact. A thing is not greater 
than a certain size because it is conceived to be so, but because it is; and 
magnitude is not infinite in virtue of increase in thought (ao8 a 16 — zz). 

Hankel and Gorland do not quote the passage about an infinite series of 
magnitudes {206 b 3—13) included in the above paraphrase; but I have 
thought that mathematicians would be interested in the distinct expression of 
Aristotle's view that the existence of an infinite series the terms of which are 
magnitudes is impossible unless it is convergent, and (with reference to 
Riemann's developments) in the statement that it does not matter to geometry 
if the straight line is not infinite in length, provided that it is as long as we 
please. 

Aristotle's denial of even the potential existence of a sum of magnitudes 
which shall exceed every definite magnitude was, as he himself implies, in 
conflict with the lemma or assumption used by Eudoxus (as we infer from 
Archimedes) to prove the theorem about the volume of a pyramid. The 
lemma is thus stated by Archimedes (Quadrature of a parabola, preface): 
" The excess by which the greater of two unequal areas exceeds the less can, 
if it be continually added to itself, be made to exceed any assigned finite 
area." We can therefore well understand why, a century later, Archimedes 
felt it necessary to justify his own use of the lemma as he does in the same 
preface ; " The earlier geometers too have used this lemma : for it is by its 
help that they have proved that circles have to one another the duplicate 
ratio of their diameters, that spheres have to one another the triplicate ratio 
of their diameters, and so on. And, in the result, each of the said theorems 
has been accepted no less than those proved without the aid of this lemma." 

Principle of continuity. 

The use of actual construction as a method of proving the existence ot 
figures having certain properties is one of the characteristics of the Elements. 
Now constructions are effected by means of straight lines and circles drawn 
in accordance with Postulates 1 — 3 ; the essence of them is that such straight 
lines and circles determine by their intersections other points in addition to 
those given, and these points again are used to determine new lines, and so on. 
This being so, the existence of such points of intersection must be postulated 
or proved in the same way as that of the lines which determine them. Yet 
there is no postulate of thfs character expressed in Euclid except Post J. 
This postulate asserts that two straight lines meet if they satisfy a certain 
condition. The condition is of the nature of a Stop«r>«5s (discrimination, or 
condition of possibility) in a problem ; and, if the existence of the point of 



i. Axx.l PRINCIPLE OF CONTINUITY 335 

intersection were not granted, the solutions of' problems in which the points of 
intersection of straight lines are used would not in general furnish the required 
proofs of the existence of the figures to be constructed. 

But, equally with the intersections of straight lines, the intersections of 
circle with straight line, and of circle with circle, are used in constructions. 
Hence, in addition to Postulate 5, we require postulates asserting the actual 
existence of points of intersection of circle with straight line and of circle 
with circle. In the very first proposition the vertex of the required equilateral 
triangle is determined as one of the intersections of two circles, and we need 
therefore to be assured that the circles will intersect. Euclid seems to assume 
it as obvious, although it is not so; and he makes a similar assumption in 
1. 2 2. It is true that in the latter case Euclid adds to the enunciation that 
two of the given straight lines must be together greater than the third ; but 
there is nothing to show that, if this condition is satisfied, the construction is 
always possible. In 1. 12, in order to be sure that the circle with a given 
centre will intersect a given straight line, Euclid makes the circle pass through 
a point on the side of the line opposite to that where the centre is. It appears 
therefore as if, in this case, he based his inference in some way upon the 
definition of a circle combined with the fact that the point within it called 
the centre is on one side of the straight line and one point of the circumference 
on the other, and, in the case of two intersecting circles, upon similar con- 
siderations. But not even in Book hi., where there are several propositions 
about the relative positions of two circles, do we find any discussion of the 
conditions under which two circles have two, one, or no point common. 

The deficiency can only be made good by the Principle of Continuity. 

Killing {Einfiihrung in die Grundlagen der Geometric, 11. p 43) gives the 
following forms as sufficient for most purposes 

(a) Suppose a line belongs entirely to a figure which is divided into two 
parts; then, if the line has at least one point common with each part, it must 
also meet the boundary between the parts; or 

{b) If a point moves in a figure which is divided into two parts, and if it 
belongs at the beginning of the motion to one part and at the end of the 
motion to the other part, it must during the motion arrive at the boundary 
between the two parts. 

I n the Questioni riguardanti le matematuhe elemeniari, l.,Art.s,pp. 123—143, 
the principle of continuity is discussed with special reference to the Postulate 
of Dedektnd, and it is shown, first, how the Postulate may be led up to and, 
secondly, how it may be applied for the purposes of elementary geometry. 

Suppose that in a segment A B of a straight line a point C determines 
two segments AC, CB. If we consider the point Cas belonging to only one 
of the two segments A C, CB, we have a division of the segment AB into two 
parts with the following properties. 

1. Every point of the segment AB belongs to one of the two parts. 

2. The point A belongs to one of the two parts (which we will call the 
first) and the point B to the other; the point C may belong indifferently to 

one or the other of the two parts according as we choose to premise. 

3. Every point of the first part precedes every point of the second in the 
order AB of the segment. 

(For generality we may also suppose the case in which the point C falls at 
A or at B. Considering C, in these cases respectively, as belonging to the 
first or second part, we still have a division into parts which have the 
properties above enunciated, one part being then a single point A or B.) 



J36 BOOK 1 [i. Axx. 

Now, considering carefully the inverse of the above proposition, we see 
that it agrees with the idea which we have of the continuity of the straight 
line. Consequently we are induced to admit as a postulate the following. 

If a segment of a straight lint AB is divided into two parts so that 
(i ) every point of the segment AB belongs to one of the parts, 

(2) the extremity A belongs to the first part and B to the second, and 

(3) any point whatever of the first part precedes any point whatever of the 
second part, in the order AB of the segment, 

there exists a point C of the segment AB {which may belong either to one 
part or to the other) such that every point of AB that precedes C belongs to the 
first part, and every point of AB that follows C belongs to the second part in 
the division originally assumed. 

(If one of the two parts consists of the single point A or B, the point C 
is the said extremity A or B of the segment.) 

This is the Postulate of Dedekind, which was enunciated by Dedekind 
himself in the following slightly different form (Stetigkcit unci irrationale Zahlen, 
187s, new edition 1905, p. 11). 

" If all points of a straight line fall into hvo classes such that every point of 
the first class lies to the left of every point of the second class, there exists one and 
only one point which produces this division of all the points into two daises, this 
division of the straight line into two parts." 

The above enunciation may be said to correspond to the intuitive notion 
which we have that, if in a segment of a straight line two points start from 
the ends and describe the segment in opposite senses, they meet in a point. 
The point of meeting might be regarded as belonging to both parts, but for 
the present purpose we must regard it as belonging to one only and subtracted 
from the other part. 

Application of Dedckind's postulate to angles. 

If we consider an angle less than two right angles bounded by two rays 
a, b, and draw the straight line connecting A, a point on a, with B, a point 
on b, we see that all points on the finite segment AB correspond univocally to 
all the rays of the angle, the point corresponding to any ray being the point 
in which the ray cuts the segment AB ; and if a ray be supposed to move 
about the vertex of the angle from the position a to the position b, the 
corresponding points of the segment AB are seen to follow in the same 
order as the corresponding rays of the angle (ab). 

Consequently, if the angle (ab) is divided into two parts so that 
(1) each ray of the angle (ab) belongs to one of the two parts, 
(t) the outside ray a belongs to the first part and the ray b to the second, 
(3) any ray whatever of the first part precedes any ray whatever of the 
second part, 
the corresponding points of the segment AB determine tteo parts of the 
segments such that 

(1) every point of the segment AB belongs to one of the two parts, 

(2) the extremity A belongs to the first part and B to the second, 

(3) any point whatever of the first part precedes any point whatever of 
the second. 

But in that case there exists a point C of AB {which may belong to one 
or the other of the two parts) such that every point of AB that precedes C 
belongs to the first part and every point of AB that follows C belongs to the 
second part. 



I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE ^37 

Thus exactly the same thing holds of c, the ray corresponding to C, with 
reference to the division of the angle {ad) into two parts. 

It is not difficult to extend this to an angle (ai) which is either flat or 
greater than two right angles; this is done (Vitali, op. cit. pp. 126—127) by 
supposing the angle to be divided into two, {ad), {di>), each less than two 
right angles, and considering the three cases in which 

(1) the ray d is such that all the rays that precede it belong to the first 
patt and those which follow it to the second part, 

(2) the ray d is followed by some rays of the first part, 

(3) the ray d is preceded by some rays of the second part. 

Application to circular arcs. 

If we consider an arc AB of a circle with centre O, the points of the arc 
correspond uni vocally, and in the same order, to the rays from the point O 
passing through those points respectively, and the same argument by which 
we passed from a segment of a straight line to an angle can be used to make 
the transition from an angle to an arc. 

Intersections of a straight line with a circle. 

It is possible to use the Postulate of Dedekind to prove that 

If a straight line has one point inside and one point outside a circle, it has 
two points common with the circle. 

For this purpose it is necessary to assume (1) the proposition with reference 
to the perpendicular and obliques drawn from a given point to a given straight 
line, namely that of all straight lines drawn from a given point to a given 
straight line the perpendicular is the shortest, and of the rest (the obliques) 
that is the longer which has the longer projection upon the straight line, while 
those are equal the projections of which are equal, so that for any given 
length of projection there are two equal obliques and two only, one on each 
side of the perpendicular, and (2) the proposition that any side of a triangle 
is less than the sum of the other two. 

Consider the circle {(7) with centre 0, and a straight line {r) with one 
point A inside and one point B outside the 
circle. 

By the definition of the circle, if R is 
the radius, 

OA<R, OB>R. 

Draw OP perpendicular to the straight 
line r. 

Then OP< OA, so that OP is always 
less than R, and P is therefore within the 
circle C. 

Now let us fix our attention on the finite segment AB of the straight 
line r. It can be divided into two parts, (1) that containing all the points H 
for which OIf< R (i.e. points inside C), and (2) that containing all the 
points K for which OK £ R (points outside C or on the circumference of C). 

Thus, remembering that, of two obliques from a given point to a given 
straight line, that is greater the projection of which is greater, we can assert 
that all the points of the segment PB which precede a point inside C are 
inside C, and those which follow a point on the circumference of C or outside 
C are outside C. 

Hence, by the Postulate of Dedekind, there exists on the segment PB a 




238 



BOOK I 



[1. Axx. 



point M such that all the points which precede it belong to the first part and 
those which follow it to the second part. 

I say that M is common to the straight line r and the circle C, or 

OM=R. 

For suppose, e.g., that OM <.R. 

There will then exist a segment (or length) a less than the difference 
between R and OM. 

Consider the point Af, one of those which follow M, such that MAT is 
equal to a. 

Then, because any side of a triangle is less than the sum of the Other two, 
OM' < OM+ MM'. 

But OM+ MM' = OM+<r<R, 

whence OM' < R, 

which is absurd. 

A similar absurdity would follow if we suppose that OM > R. 

Therefore OM mist be equal to R. 

It is immediately obvious that, corresponding to the point Mor\ the segment 
PS which is common to r and C, there is another point on r which has the 
same property, namely that which is symmetrical to M with respect to P. 

And the proposition is proved. 

Intersections of two circles. 

We can likewise use the Postulate of Dedekind to prove that 

If in a given plane a circle C has one paint X inside and one point Y outside 
another circle C ', the two circles intersect in two points. 

We must first prove the following 

Lemma. 

If O, 0' are the centres of two circles C, C, and R, R' their radii 
respectively, the straight line 00' meets the circle C in two points A, S, one 
of which is inside C" and the other outside it. 

Now one of these points must fall (r) on the prolongation of 00 
beyond O or {2) on 00 itself or {3) on the 
prolongation of OO' beyond 0. 

(1) First, suppose A to lie on OO pro- 
duced. 

Then A0=AO+ 00' = JP+ OO" (a). 

But, in the triangle OO Y, 

0Y<OY+O0, 
and, since 0Y>R\ OY=R, 
R<R+ Off. 
It follows from (a) that A0>R; and 
therefore lies outside C". 

(2) Secondly, suppose A to lie on 
00. 

Then 00 = OA + A0 = R + A0 ...($). 
From the triangle 00 X we have 

O0<OX+0X, 
and, since OX-R, 0X<R, it follows 
that 

00'<R + R, 
whence, by {£), A0 < R, so that A lies inside C 




i. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 



2 39 




Consider 



{3) Thirdly, suppose A to lie on 00 produced. 

Then R = OA = 00 + O'A (7). 

And, in the triangle 00 X, 

ox<oa+0x, w 

that is R < 00 + OX, B f 

whence, by (y), 

00 + O'A < Off + 0X, 
or O'A < 0X, 

and A lies inside C. 

It is to be observed that one of the two points A, B is in the position of 
case (1) and the other in the position of either case {2) or case (3) : whence 
we must conclude that one of the two points A, B is inside and the other 
outside the circle C. 

Proof of theorem. 

The circle C is divided by the points A, B into two semicircles, 
one of them, and suppose it to be 
described by a point moving from A 
to B. 

Take two separate points P, Q 
on it and, to fix our ideas, suppose 
that P precedes Q. 

Comparing the triangles O0P, 
OaQ, we observe that one side 00 
is common, OP is equal to OQ, and 
the angle PO0 is less than the angle 

QO0. 

Therefore 0P< 0Q. 

Now, considering the semicircle APQB as divided into two parts, so that 
the points of the first part are inside the circle C, and those of the second 
part on the circumference of C or outside it, we have the conditions necessary 
for the applicability of the Postulate of Dedekind (which is true for arcs of 
circles as for straight lines) ; whence there exists a point M separating the two 
parts. 

I say that 0M=R". 

For, if not, suppose 0M < R 1 . 

If then a signifies the difference between R' and 0M, suppose a point if, 
which follows M, taken on the semicircle such that the chord MM' is not 
greater than cr (for a way of doing this see below). 

Then, in the triangle 0MM', 

0AT < OM+ MM' < 0M+ <r, 
and therefore 0M' < R. 

It follows that M', a point on the arc MB, is inside the circle C": 
which is absurd. 

Similarly it may be proved that 0M is not greater than R. 

Hence 0M=R. 

[To find a point M' such that the chord MM' is not greater than a, we 
may proceed thus. 

Draw from M a straight line MP distinct from OM, and cut off MP on it 
equal to er/a. 




a4o BOOK I [i. Axx. 

Join OP, and draw another radius OQ such that the angle POQ is equal 
to the angle MOP. q 

The intersection, M\ of 0@ with the 
circle satisfies the required condition. 

For MM' meets OP at right angles 
in S. 

Therefore, in the right-angled triangle 
MSP, MS is not greater than MP (it is 
less, unless MP coincides with MS, when 
it is equal). 

Therefore MS is not greater than <rjt, so that MM' is not greater than <r.\ 

















BOOK I. PROPOSITIONS. 
Proposition i. 

On a given finite straight line to construct an equilateral 
triangle. 

Let AB be the given finite straight line. 
Thus it is required to con- 
s struct an equilateral triangle on 
the straight line AB. 

With centre A and distance 
AB let the circle BCD be 
described ; [Post. 3] 

ic again, with centre B and dis- 
tance BA let the circle ACE 
be described ; [Post 3] 

and from the point C, in which the circles cut one another, to 
the points A, B let the straight lines CA, CB be joined. 

[Post. 1] 

ts Now, since the point A is the centre of the circle CDB, 
AC is equal to AB. [Def. 15] 

Again, since the point B is the centre of the circle CAB, 
BC is equal to BA. [Def. 15] 

But CA was also proved equal to AB ; 

so therefore each of the straight lines CA, CB is equal to AB. 

And things which are equal to the same thing are also 

equal to one another ; [C. JV, 1] 

therefore CA is also equal to CB. 
Therefore the three straight lines CA, AB, BC are 
aj equal to one another. 



343 BOOK I [l. i 

Therefore the triangle ABC is equilateral ; and it has 
been constructed on the given finite straight line AB. 

(Being) what it was required to do. 

I. On ■ given finite straight line. The Greek usage differs from ours in that the 
definite article is employed in such a phrase as this where we have Lhe indefinite, #ri rfli 
&ot)(taw ttffttat TtrepMnfrw, "on the given finite straight line," i.e. the linite straight line 
which we choose to take. 

-3. Let AB be the given finite straight line. To be strictly literal we should have to 
translate in the reverse order "let the given finite straight line be the (straight line) AB"; 
but this order is inconvenient in other cases where there is more than one datum, e.g. in the 
sitting-out of I. 1, "let the given point be A, and the given straight Line BC," the awkward- 
ness arising from the omission of the verb in the second clause. Hence I have, for clearness' 
sake, adopted the other order throughout the book. 

S. let the circle BCD be described. Two things are here to be noted, (1) the elegant 
and practically universal use of the perfect passive imperative in constructions, ycyp&pt* 
meaning of course "let it havt ban described" or "suppose it described," ft) the impossi- 
bility of expressing shortly in a translation the force of the words in their original order. 
Ktnihet yryai$8vi BTA means literally "let a circle have been described, the (circle, namely, 
which I denote by) BCD" Similarly we have lower down " let straight lines, (namely) the 
(straight Lines) CA, CB, be joined," erffftfx&Mrai' eidtiot til PA, PB. There seems to be 
no practicable alternative, in English, hut to translate as 1 have done in the text. 

13. from the point C„.« Euclid is careful to adhere to the phraseology of Postulate I 
except that he speaks of "joining" {tTi&$x8<#rar) instead of "drawing (yp&fair). He 
does not allow himself to use the shortened expression " let the straight line FC be joined" 
(without mention of the points F, C\ until 1. 5. 

10. each of the straight lines CA, CB, irmipa tut TA, FB and 14. the three 
straight lines CA, AB, BC, td rpcU a! 1'A, AB, Br. t have, here and in all simitar 
expressions, inserted the words "straight Lines " which are not in the Greek. The possession 
of the inflected definite article enables the Greek to omit the words, but this it not possible 
in English, and it would scarcely be English to write "each of CA, CB " or "the three CA, 
AB, BC." 

It is a commonplace that Euclid has no right to assume, without pre- 
mising some postulate, that the two circles will meet in a point C. To 
supply what is wanted we must invoke the Principle of Continuity (see note 
thereon above, p. 235). It is sufficient for the purpose of this proposition and 
of 1. 21, where there is a similar tacit assumption, to use the form of postulate 
suggested by Killing. " If a line [in this case e.g. the circumference A CE] 
belongs entirely to a figure [in this case a plane] which is divided into two parts 
[namely the part enclosed within the circumference of the circle BCD and 
the part outside that circle], and if the line has at least one point common with 
each part, it must also meet the boundary between the parts [i.e. the circum- 
ference ACE must meet the circumference BCD]." 

Zeno's remark that the problem is not solved unless it is taken for granted 
that two straight lines cannot have a common segment has already been 
mentioned (note on Post a, p. 100). Thus, if AC, BC meet at F before 
reaching C, and have the part EC common, the triangle obtained, namely 
FAB, will not be equilateral, but PA, EB will each be less than AB. But 
Post 2 has already laid it down that two straight lines cannot have a common 
segment. 

Proclus devotes considerable space to this part of Zeno's criticism, but 
satisfies himself with the bare mention of the other part, to the effect that it. 
is also necessary to assume that two circumferences (with different centres) 
cannot have a common part That is, for anything we know, there may be 
any number of points C common to the two circumferences ACE, BCD. It 
is not until in. 10 that it is proved that two circles cannot intersect in more 



...] 



PROPOSITION i 



*43 



points than two, so that we are not entitled to assume it here. The most we 
can say is that it is enough foi the purpose of this proposition if one equilateral 
triangle can be found with the given base ; that the construction only gives 
two such triangles has to be left over to be proved subsequently, And indeed 
we have not long to wait ; for i. 7 clearly shows that on either side of the 
base AB only one equilateral triangle can be described. Thus 1. 7 gives us 
the number of solutions of which the present problem is susceptible, and it 
supplies the same want in 1. 22 where a triangle has to be described with 
three sides of given length ; that is, 1. 7 furnishes us, in both cases, with one 
of the essential parts of a complete Stopiff/ios, which includes not only the 
determination of the conditions of possibility but also the number of solutions 
{ir&rax&s tyxvp't, Proclus, p. so*, 5). This view of 1. 7 as supplying an 
equivalent for 111. ro absolutely needed in 1. 1 and 1. 22 should serve to correct 
the idea so common among writers of text-books that I- 7 is merely of use as a 
lemma to Euclid's proof of 1. 8, and therefore may be left out if an alternative 
proof of that proposition is adopted. 

Agreeably to his notion that it is from 1. 1 that we must satisfy ourselves 
that isosceles and scalene triangles actually exist, as well as equilateral triangles, 
Proclus shows how to draw, lirst a particular isosceles triangle, and then a 
scalene triangle, by means of the figure of the proposition. To make an 
isosceles triangle he produces AB in both directions to meet the respective 
circles in D, E, and then describes 
circles with A, B as centres and AE, 
BD as radii respectively. The result is 
an isosceles triangle with each of two 
sides double of the third side. To make 
an isosceles triangle in which the equal 
sides are not so related to the third side 
but have any given length would require 
the use of t. 3 ; and there is no object in 
treating the question at all in advance of 
1. 22. An easier way of satisfying our- 
selves of the existence of some isosceles 
triangles would surely be to conceive any 
two radii of a circle drawn and their 
extremities joined. 

There is more point in Proclus' construction of a. scalene triangle, Suppose 
AC to be a radius of one of the two 
circles, and D a point on AC lying; in 
that portion of the circle with centre A 
which is outside the circle with centre B. 
Then, joining BD, as in the figure, we 
have a triangle which obviously has all its 
sides unequal, that is, a scalene triangle. 

The above two constructions appear in 
an-NairlzI's commentary under the name 
of Heron; Proclus does not mention his 
source. 

In addition to the above construction 
for a scalene triangle (producing a triangle in which the "given" side is 
greater than one and less than the other of the two remaining sides), Heron 
has two others showing the other two possible cases, in which the " given " 
side is (r) less than, (2) greater than, either of the other two sides. 




' 





»44 BOOK I [i. i 

Proposition 2. 

To place at a given point (as an extremity) a straight line 
equal to a given straight line. 

Let A be the given point, and BC the given straight line. 
Thus it is required to place at the point A (as an extremity) 
l a straight line equal to the given 
straight line BC. 

From the point A to the point B 
let the straight line AB be joined ; 

[Post 1] 
and on it let the equilateral triangle 
10 DAB be constructed. [1. i\ 

Let the straight lines A£, BF be 
produced in a straight line with DA, 
DB; [Post a] 

with centre B and distance BC let the 
■i circle CGH be described ; [Post 3] 

and again, with centre D and distance DG let the circle GKL 
be described, [Post. 3] 

Then, since the point B is the centre of the circle CGH, 
BC is equal to BG. 
20 Again, since the point D is the centre of the circle GKL, 
DL is equal to DG. 
And in these DA is equal to DB ; 

therefore the remainder AL is equal to the remainder 
BG. [C.N. 3] 

2 S But BC was also proved equal to BG ; 

therefore each of the straight lines AL, BC is equal 
XoBG. 
And things which are equal to the same thing are also 
equal to one another ; [CJK 1] 

p therefore AL is also equal to BC. 

Therefore at the given point A the straight line AL is 
placed equal to the given straight line BC. 

(Being) what it was required to do. 

I . (as an extremity}. I have inserted these words because " to place a straight line 
at a given point " (rftht t£ Bo$4m ffij^ff^) is not quite clear enough, at least in English. 

II. Let the straight lines AB, BF be produced.,.. It will be observed that in this 
first application of Postulate a, and again in I. 5, Euclid speaks of the conJinuaiiffft of the 
straight line as that which is produced in such cases, inlikftXIja&waw and TpoaiKfttffMir#trtar 
meaning little more than dram'nr straight lines " in a straight line with " the given straight 
lines. The first place in which Euclid uses phraseology exactly corresponding to ours when 



I. a] PROPOSITION a *45 

speaking of a straight Line being produced ia in I. l6 : "let one side of it, BC t be produced 
to D " (rp<mt;&ep\1iir8u airoC >t(a irXtupA ij Br * W to A). 

13. the remainder AL...the remainder BG. Tbe Greek expressions ate X«r^ ^ 
A A and XorviJ r£ BH> and the literal translation would be 'Mi (or BG) rtmmning™ 
bat tbe shade of meaning conveyed by the position of the definite article can hardly be 
expressed in English. 

This proposition gives Proclus an opportunity, such as the Greek 
commentators revelled in, of distinguishing a multitude of cases. After 
explaining that those theorems and problems are said to have eases which 
have the same force, though admitting of a number of different figures, and 
preserve the same method of demonstration while admitting variations of 
position, and that cases reveal themselves in the etms/rue/tbn, he proceeds to 
distinguish the cases in this problem arising from the different positions 
which the given point may occupy relatively to the given straight line. It may 
be (he says) either (i) outside the line or (a) on the line, and, if (1), it may be 
either (a) on the line produced or (6) situated obliquely with regard to it ; if 
(a), it may be either (a) one of the extremities of the line or (A) an intermediate 
point on it. It will be seen that Proclus' anxiety to subdivide leads him to 
give a "case," (a) (a), which is useless, since in that "case" we are given 
what we are required to rind, and there is really no problem to solve. As 
Savile says, " qui quaerit ad punctum ponere rectam aequaiem rjj fiy rectae, 
quaerit quod datum est, quod nemo faceret nisi forte msaniat," 

Proclus gives the construction for (a) (i) following Euclid's way of taking 
G as the point in .which the circle with centre B intersects DB produced, and 
then proceeds to " cases," of which there are still more, which result from the 
different ways of drawing the equilateral triangle and of producing its sides. 

This last class of "cases" he subdivides into three according as AB is 
(1) equal to, {a) greater than or (3) less than BC, Here again "case "(i) serves 
no purpose, since, if AB is equal to BC, the problem is already solved. But 
Proclus' figures for the other two cases are worth giving, because in one of 
them the point G is on BD produced beyond D, and in the other it lies on 
BD itself and there is no need to produce any side of the equilateral triangle. 





A glance at these figures will show that, if they were used in the proposition, 
each of them would require a slight modification in the wording (r) of the 
construction, since BD is in one case produced beyond D instead of B and 
in the other case not produced at all, (a) of the proof, since BG, instead of 
being the difference between DG and DB, is in one case the sum of DG and 
DB and in the other the difference between DB and DG. 



14<> BOOK I [i. a, 3 

Modern editors generally seem to classify the cases according to the 
possible variations in the construction rather than according to differences in 
the data. Thus Lardner, Potts, and Todhunter distinguish eight cases due 
to the three possible alternatives, (1) that the given point may be joined to 
either end of the given straight line, {2) that the equilateral triangle may then 
be described on either side of the joining line, and {3) that the side of the 
equilateral triangle which is produced may be produced in either direction. 
(But it should have been observed that, where AB is greater than BC, the 
third alternative is between producing DB and not producing it at all.) Potts 
adds that, when the given point lies either on the line or on the line produced, 
the distinction which arises from joining the two ends of the line with the 
given point no longer exists, and there are only four cases of the problem 
(I think he should rather have said solutions). 

To distinguish a number of cases in this way was foreign to the really 
classical manner. Thus, as we shall see, Euclid's method is to give one case 
only, for choice the most difficult, leaving the reader to supply the rest for 
himself. Where there was a real distinction between cases, sufficient to 
necessitate a substantial difference in the proof, the practice was to give 
separate enunciations and proofs altogether as we may see, e.g., from the 
Cenia and the De section* rationis of Apollonius. 

Proclus alludes, in conclusion, to the error of those who proposed to solve 
1. 2 by describing a circle with the given point as centre and with a distance 
equal to BC, which, as he says, is a petitio principii. De Morgan puts the 
matter very clearly {Supplementary Remarks on the first six Books* of Euclid's 
Elements in the Companion to the Almanac, 1849, p. 6). We should "insist," 
he says, "here upon the restrictions imposed by the first three postulates, 
which do not allow a circle to be drawn with a compass-carried distance; 
suppose the compasses to dose of themselves the moment they cease to touch 
the paper. These two propositions [1. 2, 3] extend the power of construction 
to what it would have been if all the usual power of the compasses had been 
assumed ; they are mysterious to all who do not see that postulate iii does 
not ask for every use of the compasses." 

Proposition 3. 

Given two unequal straight lines, to cut off from the 
greater a straight line equal to the 
less. c 

Let AB, C be the two given un- 
equal straight lines, and let AB be 
the greater of them. 

Thus it is required to cut off from 
AB the greater a straight line equal 
to C the less. 

At the point A let AD be placed 
equal to the straight line C ; [1. 2] 
and with centre A and distance AD let the circle DEF be 
described. [Post 3] 




I. 3, 4] PROPOSITIONS 2—4 *4T 

Now, since the point A is the centre of the circle DEF, 

AE is equal to AD. [Def. 15] 

But C is also equal to AD. 

Therefore each of the straight lines AE, C is equal to 
AD ; so that A E is also equal to C. [C.N. 1] 

Therefore, given the two straight lines AB, C, from AB 
the greater AE has been cut off equal to C the less. 

(Being) what it was required to do. 

P roc his contrives to make a number of "cases" out of this proposition 
also, and gives as many as eight figures. But he only produces this variety by 
practically incorporating the construction of the preceding proposition, instead 
of assuming it as we are entitled to do. If Prop, 2 is assumed, there is really 
only one " case " of the present proposition, for Potts distinction between two 
cases according to the particular extremity of the straight line from which the 
given length has to be cut off scarcely seems to be worth making. 

Proposition 4. 

If two triangles have the two sides equal to two sides 
respectively, and have the angles contained by the equal straight 
lines equal, they will also have the base equal to the base, the 
triangle will be equal to the triangle, and the remaining angles 
s will be equal to the remaining angles respectively, namely those 
which the equal sides subtend. 

Let ABC, DEF be two triangles having the two sides 
AB, AC equal to the two sides DE, DF respectively, namely 
AB to DE and AC to DF, and the angle BA C equal to the 
10 angle EDF. 

I say that the base BC is also equal to the base EF, the 
triangle ABC will be equal to the triangle DEF, and the 
remaining angles will be equal to the remaining angles 
respectively, namely those which the equal sides subtend, that 
is is, the angle ABC to the angle DEF, and the angle ACB 
to the angle DEE. 

For, if the triangle ABC be 
applied to the triangle DEF, 
and if the point A be placed 
ao On the point D 

and the straight line AB 
on DE, 
then the point B will also coincide with E, because AB is 
equal to DE. 





348 BOOK I fjfc 4 

»j Again, AB coinciding with DE, 

the straight line AC will also coincide with DF, because the 

angle SAC is equal to the angle EDF; 

hence the point C will also coincide with the point F, 

because AC is again equal to DF. 
30 But B also coincided with E ; 

hence the base BC will coincide with the base EF. 

[For if, when B coincides with E and C with F, the base 

BC does not coincide with the base EF, two straight lines 

will enclose a space : which is impossible. 

35 Therefore the base BC will coincide with 

EF~\ and will be equal to it. \C.N. 4] 

Thus the whole triangle ABC will coincide with the 
whole triangle DEF, 

and will be equal to it. 

40 And the remaining angles will also coincide with the 
remaining angles and will be equal to them, 

the angle ABC to the angle DEF, 

and the angle ACB to the angle DFE, 

Therefore etc. 
*S (Being) what it was required to prove. 

1 — 3. It is a fact that Euclid's enunciations not infrequently leave something to be 
desired in point of clearness and precision. Here he speaks of the triangles having *' the 
angle equal to the angle, namely the angle contained by the equal straight lines " [rrtr yuriar 
r£ ywif tar}* §xv T h r v*& ruv law iijdtiutv wtpi.txt>M yr l v )i only one of the two angles being 
described in the latter expression (in the accusative], and a similar expression in the dative 
being left to be understood of the other angle. It is curious too that, after mentioning two 
"sides" he speaks of the angles contained by the equal "straight lints" not "sides. It 
may be that he wished to adhere scrupulously, at the outset, to the phraseology of the 
definitions, where the angle is the inclination to one another of two lines or straight lints. 
Similarly in the enunciation of I: £ he speaks of producing the equal " straight lines" as if to 
keep strictly to the wording of Postulate 1. 

t. respectively. I agree with Mr H. M. Taylor {Euclid, p. ix) that it is best to 
abandon the traditional translation of [ ' each to each," which would naturally seem to imply 
that all the four magnitudes are equal rather than (as the Greek itaripa. 1 tariff does) that 
one is equal to one and the other to the other. 

3. the base. Here we have the word bast used for the first time in the Elements. 
Proclus explains it (p. 136, 13 — tj) as meaning (1), when no side of a triangle has been 
mentioned before, the side " which is on a level with the sight " (rV rpbi tq ftipet ntifiirjjr), 
and (1), when two sides have already been mentioned, the third side. Proclus thus avoids 
the mistake made by some modern editors who explain the term exclusively with reference to 
the case where two sides have been mentioned before. That this is an error is proved (t) by 
the occurrence of .the term in the enunciations of 1. 37 etc. about triangles on the same base 
and equal bases, (1) by the application of the same term to the bases of parallelograms in 
h 35 etc. The truth is that the use of the term must have been suggested by the practice of 
drawing the particular side horizontally, as it were, and the rest of the figure above it. The 
bait of a figure was therefore spoken of, primarily, in the same sense as the base of anything 



r. 4] PROPOSITION 4 *49 

else. e.g. of a pedestal or column ; but 'vhen, as in ]. 5. two triangles were compared 
occupying other thun the norma! positions which gave rise to the name, and when two side' 
had been previously mentioned , the base was as Prod us says, necessarily the third side. 
6. subtend. Owvrttteuf br6, " to stretch under," with accusative* 
9. the angle BAC. The full Greek expression would be it irro rwp BA, AT vtpuxo^rn 
yaivtu, " the angle contained by the (straight lines) BA, AC." But it was a common practice 
of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books x.— xiij.), to 
use the abbreviation at BAl' for at BA, AT, "the (straight lines} BA, AC." Thus, on 
TtpttX'tfni being dropped, the expression would become first ^ iiri rat BAr -yafta, then 
i M BAP yttrln, and finally i brb BAr, without ywla, as we regularly find it in Euclid. 

17. if the triangle be applied to..., 13. coincide. The difference between the 
technical use of the passive t>apjt4f»T0ai " to be applitd (to)," and of the active t£np^f«> 
"to coincide (with} has been noticed above (note on Common tfetien 4, pp. 114 — j). 

J j. [For if, when B coincides... j'>. coincide with EF]. Heiberg (ParaHpumcna i» 
lid in Hermts, xxxvin., 1003, p. 56] has pointed out, as a conclusive reason for regarding 
these words as an early interpolation, that the text of an-NairM [Codex Ltidtnsis 300, 1, ed. 
Besthom- Heiberg, p. 55) does not give the words in this place but after the conclusion q.e.d., 
which shows that they constitute a scholium only. They were doubt less added b? some 
commentator who thought it necessary to explain the immediate inference that, since B 
coincides with E and C with F. the straight line BC coincides with the straight line EF, 
an inference which really follows from the definition of a straight tine and Post. 1 ; and no 
doubt the Postulate that "Two straight lines cannot enclose a space" (afterwards placed 
among the Common Notions) was interpolated at the same time. 

44. Therefore etc. Where (as here) Euclid's conclusion merely repeats the enunciation 
word for word, I shall avoid the repetition and write " Therefore etc" simply. 

In the note on Common Notion 4 I have already mentioned that Euclid 
obviously used the method of superposition with reluctance, and I have given, 
after Veronese for the most part, the reason for holding that that method is 
not admissible as a theoretical means of proving equality, although it may be 
of use as a practical test, and may thus furnish an empirical basis on which to 
found a postulate. Mr Bertrand Russell observes {Principles of Mathematics 
I. p. 405) that Euclid would have done better to assume 1. 4 as an axiom, as 
is practically done by Hilbert (Grundtagen der Geometric, p. 9). It may be 
that Euclid himself was as well aware of the objections to the method as are 
his modem critics ; but at all events those objections were stated, with almost 
equal clearness, as early as the middle of the 16th century. Peletarius 
(Jacques Peletier) has a long note on this proposition (Jn Eudidis Elementa 
gtometrica demonttratwnunt libri sex, 1557), in which he observes that, if 
superposition of lines and figures could be assumed as a method of proof, the 
whole of geometry would be full of such proofs, that it could equally well have 
been used in 1. 2, 3 (thus in 1. * we could simply have supposed the line taken 
up and placed at the point), and that in short it is obvious how far removed the 
method is from the dignity of geometry. The theorem, he adds, is obvious in 
itself and does not require proof ; although it is introduced as a theorem, it 
would seem that Euclid intended it rather as a definition than a theorem, " for 
I cannot think that two angles are equal unless I have a conception of what 
equality of angles is." Why then did Euclid include the proposition among 
theorems, instead of placing it among the axioms ? Peletarius makes the best 
excuse he can, but concludes thus : " Huius itaque propositionis veritatem non 
aliunde quam a communi iudirio petemus : cogitabimusque figuras figuris 
superponere, Mechantcum quippiam esse : intelligere verb, id demum esse 
Mathematicum." 

Expressed in terms of the modern systems of Congruence-Axioms referred 
to in the note on Common Notion 4, what Euclid really assumes amounts to 
the following : 

(1) On the line DE, there is a point E, on either side of D, such that AB 
is equal to DE. 



*5<> BOOK I [1.4 

(2) On either side of the ray DE there is a ray DF such that the angle 
EDF is equal to the angle BAC- 

It now follows that on DF there is a point ^ such that DF is equal 
to^C. 

And lastly (3), we require an axiom from which to infer that the two 
remaining angles of the triangles are respectively equal and that the bases are 
equal. 

I have shown above (pp. 229 — 230) that Hilbert has an axiom stating the 
equality of the remaining angles simply, but proves the equality of the bases. 

Another alternative is that of Pasch ( VorUsungen titer neuert Geometric, 
p. 109) who has the following "Gnmdsatz": 

If two figures AB and FGH are given (FGH not being contained in a 
straight length), and AB, FG are congruent, and if a plane surface be laid 
through A and B, we can specify in this plane surface, produced if necessary, 
two points C, D, neither more nor less, such that the figures ABC and ABD 
are congruent with the figure FGH t and the straight line CD has with the 
straight line AB or with AB produced one point common. 

I pass to two points of detail in Euclid's proof : 

(1) The inference that, since B coincides with E, and C with F, the 
bases of the triangles are wholly coincident rests, as expressly stated, on the 
impossibility of two straight tines enclosing a space, and therefore presents no 
difficulty. 

But (2) most editors seem to have failed to observe that at the very 
beginning of the proof a much more serious assumption is made without any 
explanation whatever, namely that, if A be placed on D, and AB on DE, the 
point B will coincide with £, because AB is equal to DE. That is, the 
converse of Common Notion 4 is assumed for straight lines. Proem s merely 
observes, with regard to the converse of this Common Notion, that it is only 
true in the case of things " of the same form " (i/toeiBi}), which he explains as 
meaning straight lines, arcs of one and the same circle, and angles " contained 
by lines similar and similarly situated" (p. 241, 3 — 8^. 

Savile however saw the difficulty and grappled with it in his note on the 
Common Notion. After stating that all straight lines with two points common 
are congruent between them (for otherwise two straight lines would enclose a 
space), he argues thus. Let there be two straight lines AB, DE, and let A be 
placed on D, and AB on DE. Then B will coincide with E. For, if not, 
let B fall somewhere short of E or beyond E ; and in either case it will follow 
that the less is equal to the greater, which is impossible. 

Savile seems to assume (and so apparently does Lardner who gives the 
same proof) that, if the straight lines be " applied," B will fall somewhere on 
DE or DE produced. But the ground for this assumption should surely be 
stated ; and it seems to me that it is necessary to use, not Postulate 1 alone, 
nor Postulate 2 alone, but both, for this purpose (in other words to assume, 
not only that two straight lines cannot enclose a space, but also that two straight 
lines cannot have a common segment). For the only safe course is to place A 
upon D and then turn AB about D until some point on AB intermediate 
between A and B coincides with some point on DE. In this position AB and 
DE have two points common. Then Postulate 1 enables us to infer that the 
straight lines coincide between the two common points, and Postulate 2 that 
they coincide beyond the second common point towards B and E. Thus the 
straight lines coincide throughout so far as both extend; and Savile's argument 
then proves that B coincides with E. 




I. 5] PROPOSITIONS 4, 5 151 

Proposition 5. 

In isosceles triangles the angles at the base are equal to one 
another, and, if the equal straight lines be produced further, 
the angles under the base will be equal to one another. 

Let ABC be an isosceles triangle having the side AB 
5 equal to the side AC; ' 

and let the straight lines BD, CE be produced further in a 
straight line with AB, AC. [Post. 2} 

1 say that the angle ABC is equal to the angle ACS, and 
the angle CBD to the angle BCE. 
10 Let a point F be taken at random 
on BD; 

from AE the greater let AG be cut off 
equal to AF the less ; [1. 3] 

and let the straight lines FC, GB be joined. 

[Post 1] 

is Then, since AF is equal to AG and 
AB to AC, 

the two sides FA, AC are equal to the 
two sides GA, AB, respectively ; 
and they contain a common angle, the angle FAG. 

10 Therefore the base FC is equal to the base GB, 
and the triangle AFC is equal to the triangle AGB, 

and the remaining angles will be equal to the remaining angles 
respectively, namely those which the equal sides subtend, 
that is, the angle ACF to the angle ABG, 

11 and the angle AFC to the angle AGB. [l 4] 
And, since the whole AF is equal to the whole AG, 

and in these AB is equal to AC, 
the remainder BF is equal to the remainder CG. 
But FC was also proved equal to GB ; 
v> therefore the two sides BF, FC are equal to the two sides 
CG, GB respectively ; 

and the angle BFC is equal to the angle CGB, 

while the base BC is common to them ; 
therefore the triangle BFC is also equal to the triangle CGB, 
(S and the remaining angles will be equal to the remaining 



3S* BOOK I [l. 5 

angles respectively, namely those which the equal sides 
subtend ; 

therefore the angle FBC is equal to the angle GCB, 

and the angle BCF to the angle CBG. 

4° Accordingly, since the whole angle ABG was proved 
equal to the angle ACF, 

and in these the angle CBG is equal to the angle BCF, 
the remaining angle ABC is equal to the remaining angle 
ACB; 
45 and they are at the base of the triangle ABC. 

But the angle FBC was also proved equal to the angle GCB ; 
and they are under the base. 
Therefore etc. Q. e. d. 

1. the equal straight lines (meaning the equal sida). Cf. note on the similar 
expression in Prop. 4, lines 1, 3. 

10, Let a point F be taken at random on BD, ettrj^tfu irl rQt BA Tv%6r miitur t& Z, 
where rvxhr a-rj^fiof means "a chance point/' 

17, the two sides FA, AC are equal to the two aides OA, AB respectively, l(So 
at ZA, AT 0url toTi HA, AB loat flair iuaripa tzartpf. Here, and in numberless later 
passages, I have inserted the word " sides" for the reason given in the note on 1- r, line 10. 
It would have been permissible to supply either "straight lines'' or "sides"; but on the 
whole M sides " seems to be more in accordance with the phraseology of I. 4, 

33. the base BC is common to them, i.e., apparently, common to the origin, as 
the arW-ur in pdtrti a&rww ttotrij can only refer to yuyLa and yuAa preceding. Simson wrote 
"and the base BC is common to the two triangles BFC, CGB , Todhunter left out these 
words as being of no use and tending to perplex a beginner. But Euclid evidently chose 
to quote the conclusion of 1. 4 exactly I the first phrase of that conclusion is that the bases 
(of the two triangles) are equal, and, as the equal bases are here the same base, Euclid 
naturally substitutes the worn "common" for '* equal." 

48. As " (Being) what it was required to prove " {or " do ") is somewhat long, 1 shall 
henceforth write the time-honoured "Q. e. D. and "<}. F_ F." for irtp titt ititfit and ortp 
Wet rotfyrat. 

According to Proclus (p. 250, 20) the discoverer of the fact that in any 
isosceles triangle the angles at the base are equal was Thales, who however 
is said to have spoken of the angles as being similar, and not as being equal. 
(Cf. ArisL De caelo iv. 4, 311 b 34 n-pos ifiolas ymvlat dxuVcrtu dxpoVcvov where 
equal angles are meant.) 

A pre-Euclidean proof of I. 5. 

One of the most interesting of the passages in Aristotle indicating differences 
between Euclid's proofs and those with which Aristotle was familiar, in other 
words, those of the text-books immediately preceding Euclid's, has reference to 
the theorem of 1. 5. The passage {Anal. Prior. 1. 34, 41 b 13 — 21) is so 
important that I must quote it in Kill. Aristotle is illustrating the fact that in 
any syllogism one of the propositions must be affirmative and universal 
{uttftJAou}. "This," he says, "is better shown in the case of geometrical 
propositions " {b> toii StaypapiMurtv), e.g. the proposition that the angles at Ike 
bast of an isosceles triangle are equal. 

" For let A, B be drawn [i.e. joined] to the centre. 




I. s] PROPOSITION s *S3 

"If, then, we assumed (i) that the angle AC [i.e. A + C] is equal to the 
angle BD [i.e. B + D\ without asserting generally 
that the angles of semicircles are equal, and again 
(a) that the angle C is equal to the angle D without 
making the further assumption that the two angles of 
all segments art equal, and if we then inferred, lastly, 
that, since the whole angles are equal, and equal 
angles are subtracted from them, the angles which 
remain, namely E, F, are equal, we should commit 
a petitio principii, unless we assumed [generally j 
that, when equals art subtracted from equals, tht 
remainders are equal.'' 

The language is noteworthy in some respects. 

(i ) A, B are said to be drawn (iyy/«Viii) to the centre (of the circle of 
which the two equal sides are radii) as if A, B were not the angular points but 
the sides or the radii themselves. (There is a parallel for this in Eucl. iv. 4.) 

(2) "The angle AC" is the angle which is the sum of A and C, and A 
means here the angle at A of the isosceles triangle shown in the figure, and 
afterwards spoken of by Aristotle as E, while C is the " mixed " angle between 
AB and the circumference of the smaller segment cut off by it. 

(3) The "angle of a. semicircle" (i.e. the "angle" between the diameter 
and the circumference, at the extremity of the diameter) and the " angle of a 
segment" appear in Euclid tn. 16 and 111. Def. 7 respectively, obviously as 
survivals from earlier text-books. 

But the most significant facts to be gathered from the extract are that in 
the text-books which preceded Euclid's " mixed " angles played a much more 
important part than they do with Euclid, and, in particular, that at least two 
propositions concerning such angles appeared quite at the beginning, namely 
the propositions that the (mixed) angles of semicircles art equal and that the two 
(mixed) angles of any segment of a circle art equal. The wording of the first 
of the two propositions is vague, but it does not necessarily mean more than 
that the two (mixed) angles in one semicircle are equal, and I know of no 
evidence going to show that it asserts that the angle of any one semicircle is 
equal to the angle of any other semicircle (of different size). It is quoted in 
the same form, " because the angles of semicircles are equal," in the Latin 
translation from the Arabic of Heron's Catopirica, Prop. 9 (Heron, Vol. 11., 
Teubner, p. 334), but it is only inferred that the different radii of one circle 
make equal "angles" with the circumference; and in the similar proposition 
of the Pseudo-Euclidean Catoptriea (Euclid, Vol. vn., p. 394) angles of the 
same sort in one circle are said to be equal " because they are (angles) of 
a semicircle." Therefore the first of the two propositions may be only a 
particular case of the second. 

But it is remarkable enough that the second proposition (that the two 
"angles of" any segment of a circle art equal) should, in earlier text-books, have 
been placed before the theorem of Eucl. 1. 5. We can hardly suppose it to 
have been proved otherwise than by the superposition of the semicircles into 
which the circle is divided by the diameter which bisects at right angles the 
base of the segment; and no doubt the proof would be closely connected with 
that of Thales' other proposition that any diameter of a circle bisects it, which 
must also (as Proclus indicates) have been proved by superposing one of the 
two parts upon the other. 

It is a natural inference from the passage of Aristotle that Euclid's proof of 




*54 BOOK I [i. s 

i. 5 was his own, and it would thus appear that his innovations as regards 
order of propositions and methods of proof began at the very threshold of the 
subject. 

Proof without producing the sides. 
In this proof, given by Proclus (pp. 148, 21—249, J 9)> & an ^ E are ta ^en 
on AS, AC, instead of on AB, AC produced, so that AD, AE&te equal. The 
method of proof is of course exactly like Euclid's, but it does not establish the 
equality of the angles beyond the base as well. 

Pappus* proof. 

Proclus (pp. 249, 20 — 250, 1 2) says that Pappus proved the theorem in a 
still shorter manner without the help of any construction whatever. 

This very interesting proof is given as follows : 

" Let ABC be an isosceles triangle, and AB equal to 
AC. 

Let us conceive this one triangle as two triangles, and let 
us argue in this way. 

Since AB is equal to AC, and AC to AB, 
the two sides AB, AC are equal to the two sides AC, AB. 

And the angle BA C is equal to the angle CAB, for it is 
the same. 

Therefore all the corresponding parts (in the triangles) are equal, namely 

BC to BC, 

the triangle ABC to the triangle ABC (i.e. ACB), 

the angle ABC to the angle ACB, 

and the angle ACB to the angle ABC, 

(for these are the angles subtended by the equal sides AB, A C. 

Therefore in isosceles triangles the angles at the base are equal." 

This will no doubt be recognised as the foundation of the alternative 
proof frequently given by modern editors, though they do not refer to Pappus. 
But they state the proof in a different form, the common method being to 
suppose the triangle to be taken up, turned over, and placed again upon itself, 
after which the same considerations of congruence as those used by Euclid in 
1. 4 are used over again. There is the obvious difficulty that it supposes the 
triangle to be taken up and at the same time to remain where it is, (Cf. 
Dodgson's humorous remark upon this, Euclid and Ail modern Rivals, p. 47.) 
Whatever we may say in justification of the proceeding (e.g. that the triangle 
may be supposed to leave a tract), it is really equivalent to assuming the 
construction (hypothetical, if you will) of another triangle equal in all respects 
to the given triangle ; and such an assumption is not in accordance with 
Euclid's principles and practice. 

It seems to me that the form given to the proof by Pappus himself is by far 
the best, for the reasons (i) that it assumes no construction of a second 
triangle, real or hypothetical, (2) that it avoids the distinct awkwardness 
involved by a proof which, instead of merely quoting and applying the result 
of a previous proposition, repeats, with reference to a new set of data, the 
process by which that result was established. If it is asked how we are to 
realise Pappus' idea of two triangles, surely we may answer that we keep to one 
triangle and merely view it in two aspects. If it were a question of helping a 
beginner to understand this, we might say that one triangle is the triangle 



i. 5, 6] PROPOSITIONS 5, 6 355 

looked at in front and that the other triangle is the same triangle looked at 
from behind ; but even this is not really necessary. 

Pappus' proof, of course, does not include the proof of the second part of 
the proposition about the angles under the base, and we should still have to 
establish this much in the same way as Euclid does. 

Purpose of the second part of the theorem. 

An interesting question arises as to the reason for Euclid's insertion of the 
second part, to which, it will be observed, the converse proposition 1. 6 has 
nothing corresponding. As a matter of fact, it is not necessary for any 
subsequent demonstration that is to be found in the original text of Euclid, 
but only for the interpolated second case of 1. 7 ; and it was perhaps not 
unnatural that the undoubted genuineness of the second part of 1. 5 convinced 
many editors that the second case of 1. 7 must necessarily be Euclid's also. 
Proctus' explanation, which must apparently be the right one, is that the 
second part of 1. 5 was inserted for the purpose of fore-arming the learner 
against a possible objection (frimturif), as it was technically called, which might 
be raised to 1. 7 as given in the text, with one case only. The objection would, 
as we have seen, take the specific ground that, as demonstrated, the theorem 
was not conclusive, since it did not cover all possible cases. From this point 
of view, the second part of 1. 5 is useful not only for 1. 7 but, according to 
Proclus, for 1. 9 also. Simson does not seem to have grasped Proclus' 
meaning, for he says : " And Proclus acknowledges, that the second part of 
Prop. 5 was added upon account of Prop. 7 but gives a ridiculous reason for 
it, 'that it might afford an answer to objections made against the 7th,' as if the 
case of the 7th which is left out were, as he expressly makes it, an objection 
against the proposition itself." 

Proposition 6. 

If in a triangle two angles be equal to one another, the 
sides which subtend the equal angles will also be equal to one 
another. 

Let ABC be a triangle having the angle ABC equal to 
the angle ACS', 

I say that the side AB is also equal to the 
side AC. 

For, if AB is unequal to AC, one of them is 
greater. 

Let AB be greater; and from AB the 
greater let DB be cut off equal to AC the less ; 

let DC be joined. 

Then, since DB is equal to AC, 
and BC is common, 

the two sides DB, BC are equal to the two sides AC, 
CB respectively ; 




*$6 BOOK I [i. 6 

and the angle DBC is equal to the angle ACB ; 

therefore the base DC is equal to the base AB, 
and the triangle DBC will be equal to the triangle ACB, 

the less to the greater : 
which is absurd. 

Therefore AB is not unequal to AC; 
it is therefore equal to it. 
Therefore etc 

Q. E. D. 

Euclid assumes that, because D is between A and B, the triangle DBC 
is less than the triangle ABC. Some postulate is necessary to justify this 
tacit assumption; considering an angle less than two right angles, say the 
angle ACB in the figure of the proposition, as a cluster of rays issuing from 
C and bounded by the rays CA f CB, and joining AB (where A, B are any 
two points on CA, CB respectively), we see that to each successive ray taken 
in the direction from CA to CB there corresponds one point on AB in which 
the said ray intersects AB, and that all the points on AB taken in order from 
A to B correspond uni vocally to all the rays taken in order from CA to 
CB, each point namely to the ray intersecting AB in the point. 

We have here used, for the first time in the Elements, the method of 
redact in ad absurdum, as to which I would refer to the section above (pp. 136, 
140) dealing with this among other technical terms. 

This proposition also, being the converse of the preceding proposition, 
brings us to the subject of 

Geometrical Conversion. 

This must of course be distinguished from the logical conversion of a 
proposition. Thus, from the proposition that alt isosceles triangles have the 
angles opposite to the equal sides equal, logical conversion would only enable 
us to conclude that some triangles with two angles equal are isosceles. Thus 
1. 6 is the geometrical, but not the logical, converse of 1. 5. On the other 
hand, as De Morgan points out (Companion to the Almanac, 1849, p. 7), 1. 6 is 
a purely logical deduction from t. 5 and 1. 18 taken together, as is 1. 19 also. 
For the general argument see the note on 1. 19. For the present proposition 
it is enough to state the matter thus. Let X denote the class of triangles 
which have the two sides other than the base equal, Y the class of triangles 
which have the base angles equal ; then we may call aan-X the class of 
triangles having the sides other than the base unequal non- Y the class of 
triangles having the base angles unequal. 

Thus we have 

aii x is r, [1. 5] 

All aon-X is non-K; [1. 18) 
and it is a purely logical deduction that 

All Y is X. [1. 6] 

According to Proclus (p. 252, 5 sqq.) two forms of geometrical conversion 
were distinguished. 

(1) The leading form {rpo^yovji.iyin). the conversion par excellence (rj xvpum 



t. 6] PROPOSITION 6 457 

awKrrpo^itf), is the complete or simple conversion in which the hypothesis 
and the conclusion of a theorem change places exactly, the conclusion of the 
theorem being the hypothesis of the converse theorem, which again establishes, 
as its conclusion, the hypothesis of the original theorem. The relation between 
the first part of 1. 5 and 1. 6 is of this character. In the former the hypothesis 
is that two sides of a triangle are equal and the conclusion is that the angles 
at the base are equal, while the converse {1. 6) starts from the hypothesis that 
two angles are equal and proves that the sides subtending them are equal. 

(2) The other form of conversion, which we may call partial, is seen 
in cases where a theorem starts from two or more hypotheses combined into 
one enunciation and leads to a certain conclusion, after which the converse 
theorem takes this conclusion in substitution for one of the hypotheses of 
the original theorem and from the said conclusion along with the rest of the 
original hypotheses obtains, as its conclusion, the omitted hypothesis of the 
original theorem, r, 8 is in this sense a converse proposition to 1. 4 ; for 1. 4 
takes as hypotheses ( 1 ) that two sides in two triangles are respectively equal, 
(a) that the included angles are equal, and proves (3) that the bases are equal, 
while 1. 8 takes {1) and (3) as hypotheses and proves (2) as its conclusion. It 
is clear that a conversion of the leading type must be unique, while there 
may be many partial conversions of a theorem according to the number of 
hypotheses from which it starts. 

Further, of convertible theorems, those which took as their hypothesis 
the genus and proved a, property were distinguished as the leading theorems 
(rponfyovptva), while those which started from the property as hypothesis 
and described, as the conclusion, the genus possessing that property were the 
converse theorems. 1. 5 is thus the leading theorem and [. 6 its converse, 
since the genus is in this case taken to be the isosceles triangle. 

Converse of second part of I. 5. 

Why, asks Proclus, did not Euclid convert the second part of I. 5 as well ? 
He suggests, properly enough, two reasons: (1) that the second part of 1. 5 
itself is not wanted for any proof occurring in the original text, but is only put 
in to enable objections to the existing form of later propositions to be met, 
whereas the converse is not even wanted for this purpose ; (2) that the converse 
could be deduced from t. 6, if wanted, at any time after we have passed 1. 13, 
which can be used to prove that, if the angles formed by producing two sides 
of a triangle beyond the base are equal, the base angles themselves are equal. 

Proclus adds a proof of the converse of the second part of 1. 5. i.e. of the 
proposition that, if the angles formed by producing two 
sides of_ a triangle beyond the base are equal, the triangle a 

is isosceles ; but it runs to some length and then only „ A 

effects a reduction to the theorem of 1. 6 as we have it. AA 

As the result of this should hardly be assumed, a better / \\ 

proof would be an independent one adapting Euclid's / ^ 

own method in 1. 6. Thus, with the construction of 1. 5, l"~^~~^\ 
we first prove by means of 1. 4 that the triangles BFC, /-^"^*^~^\ 

CGB are equal in all respects, and therefore that FC is y~ "^^ 

equal to GB, and the angle BFC equal to the angle CGB. D g 

Then we have to prove that AF, AG are equal. If they 
are not, let AF be the greater, and from FA cut off FH equal to GA. 
Join CH. 



i S 8 BOOK I [i.«,T 

Then we have, in the two triangles HFC, AGB, 

two sides HF, FC equal to two sides AG, GB 
and the angle HFC equal to the angle AGB. 

Therefore (l 4) the triangles HFC, AGB are equal. But the triangles 
BFC, CGB are also equal 

Therefore {if we take away these equals respectively) the triangles HBC, 
ACB are equal: which is impossible. 

Therefore AF, AG are not unequal. 

Hence AF'm equal to AG and, if we subtract the equals BF, CG respec- 
tively, AB is equal to A C. 

This proof is found in the commentary of an-Nairlzi (ed. Besthom-Heiberg, 
p. 61 ; ed. Curtze, p. 50). 

Alternative proofs of I. 6. 

Todhunter points out that I. 6, not being wanted till II. 4, could be 
postponed till later and proved by means of i. 26. Bisect the angle BAC 
by a straight line meeting the base at D. Then the triangles ABD, A CD 
are equal in all respects. 

Another method depending on 1. 26 is given by an-Nairlzi after that 
proposition. 

Measure equal lengths BD, CE along the sides BA, CA. 
Join BE, CD. 

Then [1. 4] the triangles DBC, ECB are equal in all 
respects ; 

therefore EB, DC are equal, and the angles BEC, CDB 
are equal. 

The supplements of the latter angles are equal [1. 13], 
and hence the triangles ABE, A CD have two angles equal respectively and 
the side BE equal to the side CD. 

Therefore [1. 26] AB is equal to AC. 

P HO POSITION 7. 

Given two straight lines constructed on a straight line 
[from its extremities) and meeting in a point, there cannot be 
constructed on the same straight line {from its extremities), 
and on the same side of it, two other straight lines meeting in 

5 another point and equal to the former two respectively, namely 
each to that which has the same extremity with it. 

For, if possible, given two straight lines AC, CB con- 
structed on the straight line AB and meeting 
at the point C, let two other straight lines 

10 AD, DB be constructed on the same straight 
line AB, on the same side of it, meeting in 
another point D and equal to the former two 
respectively, namely each to that which has 
the same extremity with it, so that CA is 

15 equal to DA which has the same extremity A with it, and 





I. 7] PROPOSITIONS 6, 7 259 

CB to DB which has the same extremity B with it ; and let 
CD be joined. 

Then, since AC is equal to AD, 

the angle A CD is also equal to the angle ADC; [1. 5] 
20 therefore the angle ADC is greater than the angle DCB ; 
therefore the angle CDB is much greater than the angle 
DCB. 

Again, since CB is equal to DB, 

the angle CDB is also equal to the angle DCB. 
25 But it was also proved much greater than it : 
which is impossible. 
Therefore etc. Q. E. D. 

1 — 6. In an English translation of the enunciation of this proposition it is absolutely 
necessary, in order to make it intelligible, to insert some words which are not in the Greek. 
The reason is partly that the Greek enunciation is itself very elliptical, and partly that some 
words used in it conveyed more meaning than the corresponding words in English do. 
Particularly is this the case with oC evaratrboorrat iri "there shall not be constructed upon," 
since evrUraaSat. is the regular word for constructing a triangle in particular. Tbus a Greek 
would easily understand avaraB-fyrarrat iri as meaning the construction of two lines forming 
a triangle on a given straight line as base; whereaa.ro "construct two straight lines on a 
straight line " is not in English sufficiently definite unless we explain that they are drawn 
from the ends of the straight line to meet at a point. I have had the less hesitation in putting 
in the words "from its extremities" because they are actually used by Euclid in the somewhat 
similar enunciation of 1. at. 

How impossible a literal translation into English is, if it is to convey the meaning of the 
enunciation intelligibly, will be clear from the following attempt to render literally: "On the 
same straight line there shall not be constructed two other straight lines equal, each to each, 
to the same two straight lines, (terminating) at different points on the same side, having the 
same extremities as the original straight lines " (irl riji a*rj)i eiBctai S60 raft airaii tiStlmt 
AXXor ivo evSttat focu inar4pa ixarlpa ov ewTaBfaairrai vpdr dWt^j rtai AXX^I OTjfuU^ i-wl f A ovrd 
pift r« a*"k ripara (x'oeai roll it Apx*)' tiltlaa). 

The reason why Euclid allowed himself to use, in this enunciation, language apparently 
so obscure is no doubt that the phraseology was traditional and therefore, vague as it was, 
had a conventional meaning which the contemporary geometer well understood. This is 
proved, I think, by the occurrence in Aristotle (Meteorologica 111, 5, 376 a j sqq.) of the very 
same, evidently technical, expressions. Aristotle is there alluding to the theorem given by 
Eutocius from Apollonius' Plane Loci to the effect that, if H, K be two fixed points and M 
such a variable point that the ratio of MH to MK is a given ratio (not one of equality), the 
locus of M is a circle. (For an account of this theorem see note on vt. 3 below.) Now 
Aristotle says "The lines drawn up from H, K in this ratio cannot be constructed to two 
different points of the semicircle A " (of ttr iri tCh HK inayb/urai ypawial tr roirip rif 
\t>yy uLr ffvffTad^aovTat rov £q> y A i^uxurXfou rpbt AWo teal A\\o a-rjfLtiov). 

If a paraphrase is allowed instead of a translation adhering as closely as possible to the 
original, Simson's is the best that could be found, since the fact that the straight lines form 
triangles on the same base is really conveyed in the Greek. Simson's enunciation is, Upon 
the same base, and on the same side of i(, there cannot lie two triangles line have their sides 
which are terminated in one extremity of the iate equal to one another, and liiewue these 
which art terminated at the other extremity. Th. Taylor (the translator of Proclus) attacks 
Simson's alteration as "indiscreet" and as detracting from the beauty and accuracy of 
Euclid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when 
Taylor says "Whatever difficulty learners may find in conceiving this proposition abstractedly 
is easily removed by its exposition in the figure," he really gives his case away. The fact is 
that Taylor, always enthusiastic over his author, was nettled by Simson's slighting remarks 
on Proclus' comments on the proposition. Simson had said, with reference to Proclus' 
explanation of the bearing of the second part of 1. % on 1. 7, that it was not "worth while 



j6o BOOK t [i. 7 

to relate his [rifles at full length," to which Taylor retorts "But Mr Simson was no 
philosopher ; and therefore the greatest part of these Commentaries must be considered by 
him as trifles, from the want of a philosophic genius to comprehend their meaning, and 
a taste superior to that of a mere mathematician, to discover their beauty and elegance." 

10. It would be natural to insert here the step "but the angle ACD is greater than the 
angle BCD. [C. N. 5.3" 

tl, much greater, literally ''greater by much" (roXX^i fielfav). Simson and those who 
follow him translate : " much mart then is the angle BDC greater than the angle BCD," 
but the Greek for this would have to be toXXv (or iroXii) jiaX \i r tm...JU(JW>. troXX^ ^aXXof, 
however, though used by Apotlonius, is not, apparently, found in Euclid or Archimedes. 

Just as in I. 6 we need a Postulate to justify theoretically the statement that 
CD falls within the angle ACB, so that the triangle DBC is less than the 
triangle ABC, so here we need Postulates which shall satisfy us as to the 
relative positions of CA, CB, CD on the one hand and of DC, DA, DB 
on the other, in order that we may be able to infer that the angle BDC is 
greater than the angle ADC, and the angle ACD greater than the angle BCD, 

De Morgan {sp. cit. p. 7) observes that 1. 7 would be made easy to 
beginners if they were first familiarised, as a common notion, with " if two 
magnitudes be equal, any magnitude greater than the one is greater than any 
magnitude less than the other." I doubt however whether a beginner would 
follow this easily ; perhaps it would be more easily apprehended tn the form 
"if any magnitude A is greater than a magnitude B, the magnitude A is 
greater than any magnitude equal to B, and (a fortiori) greater than any 
magnitude less than B." 

It has been mentioned already (note on 1. 5) that the second case of 1. 7 
given by Simson and in our text-books generally is not in the original text 
(the omission being in accordance with Euclid's general practice of giving 
only one case, and that the most difficult, and leaving the others to be worked 
out by the reader for himself). The second case is given by Proclus as the 
answer to a possible objection to Euclid's proposition, which should assert that 
the proposition is not proved to be universally true, since the proof given does 
not cover all possible cases. Here the objector is supposed to contend that 
what Euclid declares to be impossible may still be possible if one pair of lines 
lie wholly within the other pair of lines; and the second part of 1. 5 enables 
the objection to be refuted. 

If possible, let AD, DB be entirely within the triangle formed by AC, 
CB with AB, and let AC be equal to AD and BC 
to BD. 

Join CD, and produce AC, AD to E and F. 

Then, since AC is equal to AD, o( 

the triangle ACD is isosceles, 
and the angles ECD, FDC under the base are equal. 
But the angle ECD is greater than the angle BCD , 

therefore the angle FDC is also greater than the angle 

BCD. 

Therefore the angle BDC is greater by far than the angle BCD. 

Again, since DB is equal to CB, 
the angles at the base of the triangle BDC are equal, [1. 5] 

that is, the angle BDC is equal to the angle BCD. 

Therefore the same angle BDC is both greater than and equal to the angle 
BCD: which is impossible. 

The case in which D falls on AC or BC does not require proof, 




i. 7. 8] PROPOSITIONS 7. 8 *6i 

I have already referred (note on 1. r) to the mistake made by those 
editors who regard I. 7 as being of no use except to prove 1. 8. What 1. 7 
proves is that if, in addition to the base of a triangle, the length of the side 
terminating at each extremity of the base is given, only one triangle satisfying 
these conditions can be constructed on one and the same side of the given 
base. Hence not only does 1. 7 enable us to prove 1. 8, but it supplements 
1. 1 and 1. 22 by showing that the constructions of those propositions give one 
triangle only on one and the same side of the base. But for [. 7 this could 
not be proved except by anticipating in, 10, of which therefore 1. 7 is the 
equivalent for Book 1. purposes. Dodgson (Etttfid and his modern Rivals, 
pp. 194 — 5) puts it in another way. " It [l. 7] shows that, of airplane figures 
that can be made by hingeing rods together, the /Aree-sided ones (and these 
only) are rigid (which is another way of stating the fact that there cannot be 
two such figures on the same base). This is analogous to the fact, in relation 
to solids contained by plane surfaces hinged together, that any such solid is 
rigid, there being no maximum number of sides. And there is a close analogy 
between 1. 7, 8 and in. 23, 24. These analogies give to geometry much of its 
beauty, and I think that they ought not to be lost sight of." It will therefore 
be apparent how ill-advised are those editors who eliminate 1. 7 altogether and 
rely on Philo's proof for 1. 8. 

Proclus, it may be added, gives (pp. 2 68, 19 — 269, 10) another explanation 
of the retention of I. 7, notwithstanding that it was apparently only required 
for 1. 8. It was said that astronomers used it to prove that three successive 
eclipses could not occur at equal intervals of time, i.e. that the third could not 
follow the second at the same interval as the second followed the first ; and it 
was argued that Euclid had an eye to this astronomical application of the 
proposition. But, as we have seen, there are other grounds for retaining the 
proposition which are quite sufficient of themselves. 

Proposition 8. 

If two triangles have the two sides equal to two sides 
respectively, and have also the base equal to the base, they will 
also have the angles equal which are contained by the equal 
straight lines. 
s Let ABC, DBF be two triangles having the two sides 
AB, AC equal to the two sides 
DE, DF respectively, namely 
AB to DE, and AC to DF\ and 
let them have the base BC equal 
10 to the base EF ; 

I say that the angle BAC is 
also equal to the angle EDF. 

For, if the triangle ABC be 
applied to the triangle DEF, and if the point B be placed on 
15 the point E and the straight line BC on EF, 
the point C will also coincide with F, 
because BC is equal to EF. 





2 62 BOOK I [l8 

Then, BC coinciding with EF, 

BA, AC will also coincide with ED, DF; 
ao for, if the base BC coincides with the base EF, and the sides 
BA, AC do not coincide with ED, DF but fall beside them 
as EG, GF, 

then, given two straight lines constructed on a straight 

line (from its extremities) and meeting in a point, there will 

»5 have been constructed on the same straight line (from its 

extremities), and on the same side of It, two other straight 

lines meeting in another point and equal to the former 

two respectively, namely each to that which has the same 

extremity with it 

30 But they cannot be so constructed. [1. 7} 

Therefore it is not possible that, if the base BC be applied 

to the base EF, the sides BA, AC should not coincide with 

ED, DF; 

they will therefore coincide, 
35 so that the angle BAC will also coincide with the angle 
EDF t and will be equal to it. 

If therefore etc. q. e. d. 

19. BA, AC. The text has here " BA, CA." 

SI, fall be aide them. The Greek has the future, i-apoXXdfoLvf. TapaXk&m* means 
"to pass by without touching," "to miss" or "to deviate." 

As pointed out above (p. 157) 1, 3 is a par tied converse of t. 4. 

It is to be observed that in I. 8 Euclid is satisfied with proving the equality 
of the vertical angles and does not, as in 1. 4, add that the triangles are equal, 
and the remaining angles are equal respectively. The reason is no doubt (as 
pointed out by Proclus and by Savile after him) that, when once the vertical 
angles are proved equal, the rest follows from 1. 4, and there is no object in 
proving again what has been proved already. 

Aristotle has an allusion to the theorem of this proposition in Me&orokgka 
in. 3, 373 a 5 — 16. He is speaking of the rainbow and observes that, if equal 
rays be reflected from one and the same point to one and the same point, the 
points at which reflection takes place are on the circumference of a circle. 
"For let the broken lines ACB, AFB, ADB be all reflected from the point 
A to the point B (in such a way that) AC, AF, AD are all equal to one 
another, and the lines {terminating) at B, i.e. CB, FB, DB, are likewise all 
equal ; and let AEB be joined. It follows that the triangles art equal; for 
they are upon the equal (base) AEB." 

Heiberg {Mathtmatisehes tu Aristolelts, p. 18) thinks that the form of the 
conclusion quoted is an indication that in the corresponding proposition tc. 
Eucl. 1. 8, as it lay before Aristotle, it was maintained that the triangles were 
equal, and not only the angles, and "we see here therefore, in a clear example, 
how the stones of the ancient fabric were recut for the rigid structure of his 




i 8] PROPOSITION 8 a6j 

Elements. " I do not, however, think that this inference from Aristotle's 
language as to the form of the pre- Euclidean proposition is safe. Thus if we, 
nowadays, were arguing from the data in the passage of Aristotle, we should 
doubtless infer directly that the triangles are equal in all respects, quoting I 8 
alone. Besides, Aristotle's language is rather careless, as the next sentences 
of the same passage show. "Let perpendiculars," 
he says, " be drawn to AEB from the angles, CE 
from C, FE from ^and DE from D. These, then, 
are equal ; for they are all in equal triangles, and 
in one plane ; for all of them are perpendicular 
to AEB, and they meet at one point E. There- 
fore the (line) drawn (through C, F, D) will be a 
circle, and its centre (will be) E." Aristotle should 

obviously have proved that the three perpendiculars will meet at one point E 
cm AEB before he spoke of drawing the perpendiculars CE, FE, DE. 
This of course follows from their being "in equal triangles" {by means of 
EucL i. 26); and then, from the fact that tbe perpendiculars meet at one 
point on AB, it can be inferred that all three are in one plane. 

Philo's proof of I. 8. 

This alternative proof avoids the use of 1. 7, and it is elegant ; but it is 
inconvenient in one respect, since three cases have to be distinguished. 
Proctus gives the proof in the following order (pp. 266, 15 — 168, 14). 

I-et ABC, DEF be two triangles having the sides AB, A C equal to the 
sides DE, DE respectively, and the base BC equal to the base EF. 

Let the triangle ABC be applied to the triangle DEF, so that B is placed 
on E and BC on EF, but so that A falls on the opposite side of EF from D, 
taking the position G. Then C will coincide with F, since BC is equal to 
EF. 

Now FG will either be in a straight line with DF, or make an angle with 
it, and in the tatter case the angle will either be interior (™ro to Ivtos) to the 
figure or exterior (no.™ to Item). 

I. Let FG be in a straight line with 
DF. 

Then, since DE is equal to EG-, and 
DFG is a straight line, 

DEG is an isosceles triangle, and the 
angle at D is equal to the angle at G. 

['■ 5]. 

II. Let DF, FG form an angle interior to the figure. 
Let DG be joined. 
Then, since DE, EG are equal, 

the angle EDG is equal to the angle 
EGD. 

Again, since DF is equal to FG, 
the angle FDG is equal to the angle 
FGD. 

Therefore, by addition, 
the whole angle EDF is equal to the 
whole angle EGF. 



A 






i6 4 BOOK I [i. 8, 9 

III. Let DF, FG form an angle ex/trier to the figure. 

Let DG be joined. 

The proof proceeds as in the last case, 
except that subtraction takes the place of 
addition, and 

the remaining angle EDF is equal to the 
remaining angle ECF. 

Therefore in all three cases the angle 
EDF is equal to the angle EGF, that is, 
to the angle BAC. 

It wrill be observed that, in accordance with the practice of the Greek 
geometers in not recognising as an "angle" any angle not less than two right 
angles, the re-entrant angle of the quadrilateral JDEGF'm. ignored and the angle 
DFG is said to be outside the figure. 

Proposition 9. 

To bisect a given rectilineal angle. 

Let the angle BAC be the given rectilineal angle. 

Thus it is required to bisect it. 

Let a point D be taken at random on AB ; 
let AE be cut off from AC equal to AD ; [1. 3] 
let DE be joined, and on DE let the equilateral 
triangle DEF be constructed ; 
let AF be joined. 

I say that the angle BAC has been bisected by the 
straight line AF. 

For, since AD is equal to AE, 
and AF is common, 

the two sides DA, AF are equal to the two sides 
EA, AF respectively. 

And the base DF is equal to the base EF; 

therefore the angle DAF is equal to the angle EAF. 

[.. 8] 

Therefore the given rectilineal angle BAC has been 
bisected by the straight line AF. q, e. f. 

It will be observed from the translation of this proposition that Euclid 
does not say, in his description of the construction, that the equilateral triangle 
should be constructed on the side of HE opposite to A ; he leaves this to be 
inferred from his figure. There is no particular value in Proclus' explanation 
as to how we should proceed in case any one should assert that he could not 
recognise the existence of any space below DE. He supposes, then, the 
equilateral triangle described on the side of DE towards A, and hence has to 
consider three cases according as the vertex of the equilateral triangle falls 
on A, above A or below it. The second and third cases do not d'ffer 




>.«] 



PROPOSITIONS 8, 9 



265 



substantially from Euclid's. In the first case, where ADE is the. equilateral 
triangle constructed on DE, take any point F or\ AD, and from AE cut off 
AG equal to AF. Join DG, EF meeting in H\ and 
join AH. Then AH is the bisector required. 

Proclus also answers the possible objection that 
might be raised to Euclid's proof on the ground that 
it assumes that, if the equilateral triangle be described 
on the side of DE opposite to A, its vertex .f will lie 
within the angle BAC. The objector is supposed to 
argue that this is not necessary, but that F might fall 
either on one of the lines forming the angle or outside 
it altogether. The two cases are disposed of thus. 

Suppose Fxx> fall as shown in the two figures below respectively. 

Then, since FD is equal to FE, 
the angle FDE is equal to the angle FED. 

Therefore the angle CED is greater than the angle FDE ; and, in the 
second figure, a fortiori, the angle CED is greater than the angle BDE. 

But, since ADE is an isosceles triangle, and the equal sides are produced, 





the angles under the base are equal, 

i.e., the angle CED is equal to the angle BDE. 

But the angle CED was proved greater : which is impossible. 

Here then is the second case in which, in Proclus' view, the second part 
of i. 5 is useful for refuting objections. 

On this proposition Proclus takes occasion (p. 27 r, rj — 19) to emphasize 
the fact that the given angle must be rectilineal, since the bisection of any sort 
of angle (including angles made by curves with one another or with straight 
lines) is not matter for an elementary treatise, besides which it is questionable 
whether such bisection is always possible. "Thus it is difficult to say 
whether it is possible to bisect the so-called horn-like angle " (formed by the 
circumference of a circle and a tangent to it). 

Trisection of an angle. 

Further it is here that Proclus gives us his valuable historical note about 
the trisection of any acute angle, which (as well as the division of an angle in 
any given ratio) requires resort to other curves than circles, i.e. curves of the 
species which, after Geminus, he calls "mixed." "This," he says (p. 372, 
1 — 12), "is shown by those who have set themselves the task of trisecting such 
a given rectilineal angle. For Nicomedes trisected any rectilineal angle by 
means of the conchoidal lines, the origin, order, and properties of which he 
has handed down to us, being himself the discoverer of their peculiarity. 
Others have done the same thing by means of the quadratrices of Hippias 
and Nicomedes, thereby again using 'mixed' curves. But others, starting 
from the Archimedean spirals, cut a given rectilineal angle in a given ratio." 



i66 



BOOK I 



[1.9 



(a) T riscction by means of the conchoid. 

I have already spoken of the conchoid of Nicomedes {note on Def. t, 
pp. 160 — i); it remains to show how it could be used for trisecting an 
angle. Pappus explains this (iv. pp. 274 — 5) as follows. 

Let ABC be the given acute angle, and from any point A in AB draw 
A C perpendicular to BC. 




B O 

Complete the parallelogram FBCA and produce FA to a point E such 
that, if BE be joined, BE intercepts between AC and AE a length DE equal 
to twice AB. 

I say that the angle EBC is one-third of the angle ABC. 

For, joining A to G, the middle point of DE, we have the three straight 
lines AG, DG, EG equal, and the angle AGO is double of the angle A ED 
or EBC. 

But DE is double of AB ; 
therefore AG, which is equal to DG, is equal to AB. 

Hence the angle AGD is equal to the angle ABG. 

Therefore the angle ABD is also double of the angle EBC; 
so that the angle EBC is one-third of the angle ABC. 

So far Pappus, who reduces the construction to the drawing of BE so 
that DE shall be equal to twice AB. 

This is what the conchoid constructed with B as pole, AC 'as directrix, and 
distance equal to twice AB enables us to do ; for that conchoid cuts AE in 
the required point E. 

(6) Use of the quadratrix. 

The plural quadratrices in the above passage is a Hellenism for the 
singular quadratrix, which was a curve discovered by Hippias of El is about 
420 B.C. According to Prod us (p. 356, 11) Hippias proved its properties; 
and we are told (1) in the passage quoted above that Nicomedes also 
investigated it and that it was used for trisecting an angle, and (2) by Pappus 
(iv. pp. 350, 33— 353, 4) that it was used by Dinostratus and Nicomedes and 
some more recent writers for squaring the circle, whence its name. It is 
described thus (Pappus iv. p. 352). 

Suppose that ABCD is a square and BED a quadrant of a circle with 
centre A. 

Suppose (1) that a radius of the circle moves 
uniformly about A from the position AB to the 
position AD, and (*) that in the same time the 
line BC moves uniformly, always parallel to itself, 
and with its extremity B moving along BA, from 
the position BC to the position AD. 

Then the radius AE and the moving line BC 
determine at any instant by their intersection a 
point F, 

The locus of F'\s the quadratrix. 




I. 9, 10] PROPOSITIONS o, 10 167 

The property of the curve is that, if F is any point, the arc BED is 
to the arc ED as AB is to FH. 

In other words, if ^ is the angle FAD, p the radius vector AFa.nd a the 
side of the square, 

(p sin $)/« = $/Jt. 

Now the angle EAD can not only be trisected but divided in any given 
ratio by means of the quadratrix (Pappus iv. p. 386). 

For let FJfbe divided at JCin the given ratio. 

Draw KL parallel to AD, meeting the curve in L ; join AL and produce 
it to meet the circle in N. 

Then the angles EAN, NAD are in the ratio of FK to KH, as is easily 
proved. 

(e) Use of the spiral of Archimedes. 

The trisect ion of an angle, or the division of an angle in any ratio, by 
means or the spiral of Archimedes is of course an equally simple matter. 
Suppose any angle included between the two radii vectores OA and OB of the 
spiral, and let it be required to cut the angle AOB in a given ratio. Since 
the radius vector increases proportionally with the angle described by the 
vector which generates the curve (reckoned from the original position of the 
vector coinciding with the initial line to the particular position assumed), we 
have only to take the radius vector OB (the greater of the two OA, OB), 
mark off OC along it equal to OA, cut CB in the given ratio (at D say}, and 
then draw the circle with centre and radius OD cutting the spiral in E. 
Then OE will divide the angle AOB in the required manner. 

Proposition 10. 
To bisect a given finite straight line. 

Let AB be the given finite straight line. 

Thus it is required to bisect the finite straight line AB. 

Let the equilateral triangle ABC be 
constructed on it, [1. r] 

and let the angle ACB be bisected by the 
straight line CD ; ft, 9] 

I say that the straight line AB has 
been bisected at the point D. . 

For, since AC is equal to CB, 
and CD is common, 

the two sides A C, CD are equal to the two sides BC, 
CD respectively ; 

and the angle A CD is equal to the angle BCD ; 

therefore the base AD is equal to the base BD. [1. 4] 
Therefore the given finite straight line AB has been 
bisected at D. q. e, f. 




268 



BOOK I 



[>■ 



A poll on i us, we are told (P rod us, pp. 279, 16—280, 4), bisected a straight 
line AB by a construction tike that of 1. 1. 
With centres A, B, and radii AB, BA respec- 
tively, two circles are described, intersecting in 
C, p. Joining CD, AC, CB, AD, DB, Apoi- 
lonius proves in two steps that CD bisects AB. 






(1) Since, in the triangles A CD, BCD, 
two sides AC, CD are equal to two sides 
BC, CD, 
and the bases AD, BD are equal, 
the angle A CD is equal to the angle 
BCD. 




[1.8] 



(2) The latter angles being equal, and AC being equal to CB, while CE 
is common, 

the equality of AE, EB follows by I. 4. 

The objection to this proof is that, instead of assuming the bisection of 
the angle ACB, as already effected by 1. 9, Apollo nius goes a step further 
back and embodies a construction for bisecting the angle. That is, he 
unnecessarily does over again what has been done before, which is open to 
objection from a theoretical point of view. 

Proclus (pp. 277, 25 — 279, 4) warns us against being moved by this 
proposition to conclude that geometers assumed, as a preliminary hypothesis, 
that a line is not made up of indivisible parts (1$ AjttpAr). This might be 
argued thus. If a line is made up of indivisibles, there must be in a finite 
line either an odd or an even number of them. If the number were odd, 
it would be necessary in order to bisect the line to bisect an indivisible (the 
odd one). In that case therefore it would not be possible to bisect a straight 
line, if it is a magnitude made up of indivisibles. But, if it is not so made 
up, the straight line can be divided ad infinitum or without limit (hf irtipov 
faatpttTai). Hence it was argued (^acrtV), says Proclus, that the divisibility 
of magnitudes without limit was admitted and assumed as a geometrical 
principle. To this he replies, following Geminus, that geometers did indeed 
assume, by way of a common notion, that a continuous magnitude, i.e. a 
magnitude consisting of parts connected together (owrjfijtow), is divisible 
(SuuptTov). But infinite divisibility was not assumed by them ; it was proved 
by means of the first principles applicable to the case. "For when," he 
says, " they prove that the incommensurable exists among magnitudes, and 
that it is not all things that are commensurable with one another, what 
else will any one say that they prove but that every magnitude can be 
divided for ever, and that we shall never arrive at the indivisible, that 
is, the least common measure of the magnitudes? This then is matter of 
demonstration, whereas it is an axiom that everything continuous is divisible, 
so that a finite continuous line is divisible. The writer of the Elements 
bisects a finite straight line, starting from the latter notion, and not from any 
assumption that it is divisible without limit" Proclus adds that the proposition 
may also serve to refute Xenocrates' theory of indivisible lines (di-ojiu* ypapjiat). 
The argument given by Proclus to disprove the existence of indivisible lines 
is substantially that used by Aristotle as regards magnitudes generally (cf 
Physkt vi. 1, 231 a 21 sqq. and especially vl. 2, 133 b 15 — 32). 




J. n] PROPOSITIONS io, u 269 

Proposition ii. 

To draw a straight line at right angles to a given straight 
line from a given point on it. 

Let AB be the given straight line, and C the given point 
on it. 

s Thus it is required to draw from the point C a straight 
line at right angles to the straight 
line AB. 

Let a point D be taken at ran- 
dom on AC; 
10 let CE be made equal to CD ; [1. 3] 
on DE let the equilateral triangle 
FDE be constructed, [1. i] 

and let FC be joined ; 

I say that the straight line FC has been drawn at right 
is angles to the given straight line AB from C the given point 
on it. 

For, since DC is equal to CE, 
and CF is common, 

the two sides DC, CF are equal to the two sides EC, 
20 CF respectively ; 

and the base DF is equal to the base FE ; 

therefore the angle DCF is equal to the angle ECF\ 

M] 
and they are adjacent angles. 

But, when a straight line set up on a straight line makes 

»s the adjacent angles equal to one another, each of the equal 

angles is right ; [Def. 10] 

therefore each of the angles DCF, FCE is right. 

Therefore the straight line CF has been drawn at right 

angles to the given straight line AB from the given point 

30 C on it. 

Q. E. F. 

10. let CB be made equal to CD. The verb is mbrSu which, as welt as the othet 
parts of Ktitiai, a constant iy used lor the passive of rWiftu " lo plats " ; and the latter word 
it constantly used in the sense of making, e.g., one straight line equal to another straight line. 

1 )e Morgan remarks that this proposition, which is " to bisect the angle 
made by a straight line and its continuation " [i.e. a flat angle], should be a 
particular case of 1. 9, the constructions being the same. Thjs is certainly 



J7o 



BOOK I 



['■ 



worth noting, though I doubt the advantage of rearranging the propositions 
in consequence. 

Apollonius gave a construction for this proposition (see P rod us, p. 282, 8) 
differing from Euclid's in much the same way as his construction for bisecting 
a straight line differed from that of 1. 10. Instead of assuming an equilateral 
triangle drawn without repeating the process of 1. 1, Apollonius takes D and 
E equidistant from C as in Euclid, and then draws circles in the manner of 




1. 1 meeting at F This necessitates proving again that DF\s equal to FE\ 
whereas Euclid's assumption of the construction of 1. 1 in the words " let the 
equilateral triangle FDE be constructed " enables him to dispense with the 
drawing of circles and with the proof that DF is equal to FE at the same 
time. While however the substitution of Apollonius' constructions for 1. 10 
and 1 1 would show faulty arrangement in a theoretical treatise like Euclid's, 
they are entirely suitable for what we call practical geometry, and such may 
have been Apollonius' object in these constructions and in his alternative for 

'- »3- . . . . 

Proclus gives a construction for drawing a straight line at right angles to 
another straight line but from one end of it, instead of from an intermediate 
point on it, it being supposed (for the sake of argument) that we are not 
permitted to produce the straight line. In the commentary of an-Nairizi (ed. 
Besthorn-Heiberg, pp. 73 — 4; ed. Curtze, pp. 54 — 5) this construction is 
attributed to Heron. 

Let it be required to draw from A a straight line at right angles to AB. 

On AB take any point C, and in the manner of the proposition draw CE 
at right angles to AB. 

from CE cut off CD equal to AC, bisect the 
angle ACE by the straight line CF\ [1. 9] 

and draw DF at right angles to CE meeting CF 
in F. Join FA. 

Then the angle FAC will be a right angle. 

For, since, in the triangles ACF, DCF, the 
two sides AC, CF are equal to the two sides 
DC, CF respectively, and the included angles 
ACF, DCFare equal, 

the triangles are equal in all respects. [1, 4] 

Therefore the angle at A is equal to the angle at D, and is accordingly a 
right angle. 









E 






F 


D 




















A ( 




a 



PROPOSITION 12. 

To a given infinite straight tine, from a given point 
which is not on it, to draw a perpendicular straight line. 

Let AB be the given infinite straight line, and C the 
given point which is not on it ; 




I. u] PROPOSITIONS ii, 12 371 

Sthus it is required to draw to the given infinite straight 
line AB, from the given point 
C which is not on it, a per- 
pendicular straight line. 

For let a point D be taken 

10 at random on the other side of 

the straight line AB, and with 

centre C and distance CD let 

the circle EFG be described ; 

[Post. 3] 

let the straight line EG 

ij be bisected at H, [1. 10] 

and let the straight lines CG, CH, CE be joined. 

[Post. 1] 
I say that CH has been drawn perpendicular to the given 
infinite straight line AB from the given point C which is 
not on it. 
jo For, since GH is equal to HE, 
and HC is common, 

the two sides GH, HC are equal to the two sides 
EH, HC respectively ; 
and the base CG is equal to the base CE ; 
35 therefore the angle CHG is equal to the angle EHC. 

[1. 8] 

And they are adjacent angles. 

But, when a straight line set up on a straight line makes 

the adjacent angles equal to one another, each of the equal 

angles is right, and the straight line standing on the other is 

jo called a perpendicular to that on which it stands. [Def. 10] 

Therefore CH has been drawn perpendicular to the given 

infinite straight line AB from the given point C which is 

not on it 

Q. E. F. 

1. a perpendicular straight line, tiitrer ti$ttv ypapph'- This is the full expression 
for a ptrpendkutar, KiBtrm meaning >tt f<il( or Ut down, so that the expression corresponds 
10 our plumb-lint, if jcd&rar is however constantly used atone for a perpendicular, ypafi^ 
being understood. 

10. on the other side of the straight line AB, literally " towards the other parts of 
the straight line AB," irl ri Irtpa ftfpif rt,s AB. Cf. "on the same side" (6rl ri au-ri 
pif"!) in Post, s *»d "in both directions" (ty inirtpa ri nipt)) in Def. 13. 

"This problem," says Proctus (p. 183, 7 — 10), "was first investigated 
by Oenopides [5 th cent b,c], who thought it useful for astronomy. He 
however calls the perpendicular, in the archaic manner, (a line drawn) 



2j2 BOOK I [i. it 

gnomon-wise (koto yvu/iova), because the gnomon is also at right angles to the 
horizon." In this earlier sense the gnomon was a staff placed in a vertical 
position for the purpose of casting shadows and so serving as a means of 
measuring time (Cantor, Gezchichle der Mathemattk, i s , p. 161). The later 
meanings of the word as used in Eucl. Book n. and elsewhere will be 
explained in the note on Book n. Def. 2, 

Proclus says that two kinds of perpendicular were distinguished, the "plane" 
(iiriwtSos) and the "solid" (err«pea), the former being the perpendicular 
dropped on a line is a plane and the latter the perpendicular dropped on a 
plane. The term "solid perpendicular" is sufficiently curious, but it may 
perhaps be compared with the Greek term " solid locus " applied to a conic 
section, apparently on the ground that it has its origin in the section of a 
solid, namely a cone. 

Attention is called by most editors to the assumption in this proposition 
that, if only D be taken on the side of AB remote from C, the circle described 
with CD as radius must necessarily cut AB in two points. To satisfy us of 
this we need, as in I, 1, some postulate of continuity, e.g. something like that 
suggested by Killing (see note on the Principle of Continuity above, p. 235): 
"If a point [here the point describing the circle] moves in a figure which is 
divided into two parts [by the straight line J, and if it belongs at the beginning 
of the motion to one part and at another stage of the motion to the other 
part, it must during the motion cut the boundary between the two parts," and 
this of course applies to the motion in two directions from D. 

But the editors have not, as a rule, noticed a possible objection to the 
Euclidean statement of this problem which is much more difficult to dispose 
of at this stage, i.e. without employing any proposition later than this in 
Euclid's order. How do we know, says the supposed critic, that the circle 
does not cut AB in three or more points, in which case there would be not 
one perpendicular but three or more? Proclus (pp. 186, 12—489, *>) tries to 
refute this objection, and it is interesting to follow his argument, though it 
will easily be seen to be inconclusive. He takes in order three possible 
suppositions. 

1. May not the circle meet AB in a third point K between the middle 
point of GE and either extremity of it, taking the form drawn in the figure 
appended ? 

Suppose this possible. Bisect GE in H. Join CH, and produce it to 
meet the circle in L. Join CG, CK, CE. 

Then, since CG is equal to CE, and 
CH is common, while the base GH is 
equal to the base HE, 

the angles CHG, CHE are equal and, 
since they are adjacent, they are both right. 

Again, since CG is equal to CE, 
the angles at G and E are equal. 

Lastly, since CK is equal to CG and 
also to CE, the angles CGK, CKG are 
equal, as also are the angles CKE, CEK. 

Since the angles CGK, CEK are equal, it follows that 

the angles CKG, CKE are equal and therefore both right. 

Therefore the angle CKH'\% equal to the angle CHK, 
and CH is equal to CK. 





I. 12] PROPOSITION 12 273 

But CK is equal to CL, by the definition of the circle ; therefore CH is 
equal to CL : which is impossible. 

Thus Proclus; but why should not the circle meet AB in H as well as .X? 

2. May not the circle meet AB in // the middle point of GE and take 
the form shown in the second figure? 

In that case, says Proclus, join CG, CH, CE as before. Then bisect ME 
at K, join CK and produce it to meet 
the circumference at L. 

Now, since HK is equal to KE, CK 
is common, and the base CH is equal to 
the base CE, 

the angles at K are equal and therefore 
both right angles. 

Therefore the angle CHK is equal to 
the angle CKH, whence CK is equal to CH 
and therefore to CL : which is impossible. 

So Proclus ; but why should not the circle meet AB in A* as well as Hf 

3, May not the circle meet AB in two points besides G, E and pass, 
between those two points, to the side of A3 towards C, as in the next figure ? 

Here again, by the same method, Proclus proves that, K, L being the 
other two points in which the circle cuts 
AB, 

CK is equal to CH, 

and, since the circle cuts CH'm M, 

CM is equal to CK and therefore to 
CH: which is impossible. 

But, again, why should the circle not 
cut AB in the point H a& well? 

In fact, Proclus' cases are not mutually 
exclusive, and his method of proof only enables us to show that, if the circle 
meets AB in one more point besides G, E, it must meet it in more points 
still. We can always find a new point of intersection by bisecting the distance 
separating any two points of intersection, and so, applying the method ad 
infinitum, we should have to conclude ultimately that the circle with radius 
CH (or CG) coincides with AB. It would follow that a circle with centre 
C and radius greater than CH would not meet AB at all. Also, since all 
straight lines from C to points on AB would be equal in length, there would 
be an infinite number of perpendiculars from C on AB. 

Is this under any circumstances possible ? It is not possible in Euclidean 
space, but it is possible, under the Riemann hypothesis (where a straight line 
is a " closed series " and returns on itself), in the case where C is the pole of 
the straight line AB. 

It is natural therefore that, for a proof that in Euclidean space there is 
only one perpendicular from a point to a straight line, we have to wait until 
1. 16, the precise proposition which under the Riemann hypothesis is only valid 
with a certain restriction and not universally. There is no difficulty involved 
by waiting until 1. 16, since t. 12 is not used before that proposition is reached; 
and we are only in the same position as when, in order to satisfy ourselves of 
the number of possible solutions of 1. r, we have to wait till 1. 7. 

But if we wish, after all, to prove the truth of the assumption without 
recourse to any later proposition than 1. 12, we can do so by means of this 
same invaluable 1. 7. 




»74 



BOOK I 



[l, 12 




If the circle intersects AB as before in G, E, let H be the middle point of 
GE, and suppose, if possible, that the 
circle also intersects AB in any other point 
K on AH. 

From H, on the side of AB opposite to 
C, draw HL at right angles to AB, and 
make HL equal to HC. 

Join CG, EG, CK, LK. 

Now, in the triangles CHG, EHG, 
CHis equal to /.//, and HG is common. 

Also the angles CHG, EHG, being 
both right, are equal. 

Therefore the base CG is equal to the base EG. 

Similarly we prove that CK is equal to LK. 

But, by hypothesis, since K is on the circle, 

CK is equal to CG. 

Therefore CG, CK, LG, LK are all equal. 

Now the next proposition, i. 13, will tell us that CH, HL are in a straight 
line; but we will not assume this. Join CE, 

Then on the same base CE and on the same side of it we have two pairs 
of straight lines drawn from C, L to G and K such that CG is equal to CK 
and EG to LK 

But this is impossible [1. 7 \ 

Therefore the circle cannot cut BA or BA produced in any point other 
than G on that side of CL on which G is. 

Similarly it cannot cut AB or AB produced at any point other than E 
on the other side of CL. 

The only possibility le/t therefore is that the circle might cut AB in the 
same point as that in which CL cuts it. But this is shown to be impossible 
by an adaptation of the proof of I. 7, 

For the assumption is that there may be some point M on CL such thai 
CM is equal to CG and LM to LG. 

If possible, let this be the case, and produce CG 
to N. 

Then, since CM is equal to CG, 
the angle NGM is equal to the angle GML [1. 5, part 2]. 

Therefore the angle GML is greater than the angle 
MGL. 

Again, since LG k equal to LM, 
the angle GML is equal to the angle MGL. 

But it was also greater : which is impossible. 

Hence the circle in the original figure cannot cut AB in the point in 
which CL cuts it. 

Therefore the circle cannot cut AB in any point whatever except G and E. 

[This proof of course does not prove that CK is less than CG, but only 
that it is not equal to it. The proposition that, of the obliques drawn 
from C to AB, that is less the foot of which is nearer to H can only be proved 
later. The proof by 1. 7 also fails, under the Riemann hypothesis, if C, L are 
the poles of the straight line AB, since the broken lines CGL, CKL etc. 
become equal straight lines, all perpendicular to AB.] 

Proclus rightly adds (p. z8q, 18 sqq.) that it is not mcessary to take D on 
the side of AB away from A if an objector " says that there is no space on 





1. is, 13] PROPOSITIONS is, 13 »75 

that side." If it is not desired to trespass on that side of AB, we can take D 
anywhere on AB and describe the arc of a circle between D and the point 
where it meets AB again, drawing the arc on the side of AB on which C is. 
If it should happen that the selected point D is such that the circle only meets 
AB in oik point (D itself), we have only to describe the circle with CD as 
radius, then, if E be a point on this circle, take Fa point further from C than 
E is, and describe with CF as radius the circular arc meeting AB in two 
points. 



Proposition 13. 

If a straight line set up on a straight line make angles, it 
wilt make either two right angles or angles equal to two right 
angles. 

For let any straight line AB set up on the straight line 
s CD make the angles CBA, ABD j 

I say that the angles CBA, ABD 
are either two right angles or equal to 
two right angles. 

Now, if the angle CBA is equal to 
10 the angle ABD, 

they are two right angles. [Def. 10] 

But, if not, let BE be drawn from the point B at right 

angles to CD ; [1. n] 

therefore the angles CBE, EBD are two right angles. 

is Then, since the angle CBE is equal to the two angles 

CBA, ABE, 

let the angle EBD be added to each ; 
therefore the angles CBE, EBD are equal to the three 
angles CBA, ABE, EBD. [C X a] 

M Again, since the angle DBA is equal to the two angles 
DBE, EBA, 

let the angle ABC be added to each ; 
therefore the angles DBA. ABC are equal to the three 
angles DBE, EBA, ABC. [ax*] 

ij But the angles CBE, EBD were also proved equal to 
the same three angles ; 

and things which are equal to the same thing are also 
equal to one another ; [C. X 1] 

therefore the angles CBE, EBD are also equal to the 
3a angles DBA, ABC. 



376 BOOK I [i. 13, 14 

But the angles CBE, EBD are two right angles ; 

therefore the angles DBA, ABC are also equal to two 

right angles. 

Therefore etc. 

Q. E, D, 

17. let the angle EBD be added to each, literally "let the angle EBD be added 
(so as to be) common, " *a*fy rpoandeBu % irrh EBA. Similarly tow)) d^pjsrtfw is used of 
subtracting a straight line or angle from each of two others. "Let the common angle EBD 
be added is clearly an inaccurate translation, for the angle is not common before it 13 added, 
i.e. the mur^i is proleptie. "Let the common angle be tuitraclid" as a translation of rar^ 
d<t?W?l<j8u would be less unsatisfactory, it is true, but, as it is desirable to use corresponding 
words when translating the two expressions, it seems hopeless to attempt to keep the word 
"common,'' and I have therefore said " to each" and "from each. " simply. 



Proposition 14. 

If with any straight line, and at a point on it, two straight 
lines not lying on the same side make the adjacent angles equal 
to two right angles, the two straight lines will be in a straight 
line with one another. 

5 For with any straight line AB, and at the point B on it, 
let the two straight lines BC, BD not lying on the same side 
make the adjacent angles ABC, ABD equal to two right 
angles ; 

I say that BD is in a straight line with CB. 

w For, if BD is not in a straight line 
with BC, let BE be in a straight line 
with CB. 

Then, since the straight line AB 5 — 
stands on the straight line CBE, 

15 the angles ABC, ABE are equal to two right angles. 

[<■ n] 

But the angles ABC, ABD are also equal to two right angles ; 
therefore the angles CBA, ABE are equal to the angles 
CBA, ABD. [Post 4 and C. N. t] 

Let the angle CBA be subtracted from each ; 
20 therefore the remaining angle ABE is equal to the remaining 
angle ABD, [C. N. 3] 

the less to the greater : which is impossible. 
Therefore BE is not in a straight line with CB. 
Similarly we can prove that neither is any other straight 
aj line except BD. 




I. 14. IS] PROPOSITIONS 13—15 277 

Therefore CB is in a straight line with BD. 
Therefore etc. 

Q. E. D. 

i. If with any straight line... There is no greater difficulty in translating the works 
of the Greek geometers than that of accurately giving the force of prepositions, *pbi, for 
instance, is used in all sorts of expressions with various shades of meaning. The present 
enunciation begins 'EAr Tp&t Tttfi ciffcta rai tQ rpAi avrv eTtfttitp, and it is really necessary in 
this one sentence to translate rp6i by three different words, wit A, at, and en. The first rp6t 
must be translated by with because two straight lines " make" an angle with one another. On 
the other hand, where [he similar expression rpot rp bofalfy t&dcta occurs 1 in I. 33, but it is 
a question of M constructing ™ an angle (riPcrr^ffaffPat), we have to say "to construct on a 
given straight line." Againtt would perhaps be the English word coming nearest to 
expressing all these meanings of irp£t, but it would be intolerable as a translation. 

17. Todhunter points out tbat for the inference in this line Post. 4, that all right angles 
are equal, is necessary as well as the Common Notion that things which are equal to the same 
thing [or rather, here, to squat things) are equal. A similar remark applies to steps in the 
proofs of I. 15 and I. 18. 

34. we can prove. The Greek expresses this by the future of the verb, &*t%o/ur, 
"we shall prove," which however would perhaps be misleading in English. 

P rocki 5 observes (p. 297) that two straight lines on the same side of another 
straight line and meeting it in one and the same 
point may make with one and the same portion 
of the straight line terminated at the point two 
angles which are together equal to two right angles, 
in which case however the two straight lines would 
not be in a straight line with one another. And 
he quotes from Porphyry a construction for two 
such straight lines in the particular case where they 
form with the given straight line angles equal 
respectively to half a right angle and one and a 
half right angles. There is no particular value in 
the construction, which will be gathered from the annexed figure where CB, 
CF are drawn at the prescribed inclinations to CD. 

Proposition 15. 

If two straight lines cut one another, they make the vertical 
angles equal to one another. 

For let the straight lines AB, CD cut one another at the 
point E ; 
5 I say that the angle AEC is equal to i^^ 
the angle DEB, _\I_ 

and the angle CEB to the angle D 
AED. 

For, since the straight line AE stands 
10 on the straight line CD, making the angles CEA, AED, 
the angles CEA, AED are equal to two right angles 

L'- '3] 





»78 BOOK I [l tS 

Again, since the straight line DE stands on the straight 
line AB, making the angles AED, DEB, 

the angles AED, DEB are equal to two right angles. 

[«■ '3] 

«s But the angles CEA, A ED were also proved equal to 
two right angles ; 

therefore the angles CEA, A ED are equal to the 

angles AED DEE. [Post 4 and C. If. 1] 

Let the angle AED be subtracted from each ; 

*> therefore the remaining angle CEA is equal to the 

remaining angle BED. [C. If. 3] 

Similarly it can be proved that the angles CEB, DEA 

are also equal. 

Therefore etc. Q. E, D. 

*5 [Porism. From this it is manifest that, if two straight 
lines cut one another, they will make the angles at the point 
of section equal to four right angles.] 

1. the vertical angles. The difference between adjacent angles (ai ifeffit fut/ku) and 
vertical fugles (td Kara xopv<pfyv yuvttu) is t hi lis explained by Proclus (p. 398, 14—34). The 
first term describes the angles made by two straight lines when one only it divided by the 
other, i.e. when one straight Line meets another at a point which is not either of its extremi- 
ties, but is not itself produced beyond the point of meeting. When the first straight line is 
produced, so that the lines cross at the point, they make two pairs of vertical angles (which 
are more clearly described as vertically opposite angles), and which are so called because their 
convergence is from opposite directions to one point (the intersection of the lines) as vertex 

16. at the point of section, literally " at the section," rpii tjj 7-0^5, 

This theorem, according to Eudemus, was first discovered by Thales, but 
found its scientific demonstration in Euclid (Proc)us, p. 2991, 3 — 6). 

Proclus gives a converse theorem which may be stated thus. If a straight 
line is met at one and the same point intermediate in its length by two other 
straight lines on different sides of it and such as to make the vertical angles 
equal, the latter straight lines are in a straight line with one another. The 
proof need not be given, since it is almost self-evident, whether (1) it is direct, 
by means of 1. 13, 14, or (2) indirect, by reductio ad abmrdum depending 
on 1. 15. 

The balance of ms. authority seems to be against the genuineness of this 
Porism, but Proclus and Psellus both have it. The word is not here used, as it 
is in the title of Euclid's lost Potisms, to signify a particular class of independent 
propositions which Proclus describes as being in some sort intermediate between 
theorems and problems (requiring us, not to bring a thing into existence, but 
to find something which we know to exist). Porism has here (and wherever 
the term is used in the Elements) its second meaning ; it is what we call a 
corollary, i.e. an incidental result springing from the proof of a theorem or the 
solution of a problem, a result not directly sought but appearing as it were by 
chance without any additional labour, and constituting, as Proclus says, a sort 
of windfall (ippaiov) and bonus (ntpSos). These Porisms appear in both the 



i. 15, i6] PROPOSITIONS 15, 16 179 

geometrical and arithmetical Books of the Elements, and may either result 
from theorems or problems. Here the Porism is geometrical, and springs out 
of a theorem ; vit. 1 affords an instance of an arithmetical Porism. As an 
instance of a Porism to a problem Proclus cites " that which is found in the 
second Book" (ri b t^ Stvtipif jSt^Aup mi^mfov); but as to this see notes on 
11. 4 and iv. 15. 

The present Porism, says Proclus, formed the basis of " that paradoxical 
theorem which proves that only the following three (regular) polygons can fill 
up the whole space surrounding one point, the equilateral triangle, the square, 
and the equilateral and equiangular hexagon." We can in fact place round a 
point in this manner six equilateral triangles, three regular hexagons, or four 
squares. "But only the angles of these regular figures, to the number specified, 
can make up four right angles : a theorem due to the Pythagoreans." 

Proclus further adds that it results from the Porism that, if any number of 
straight lines intersect one another at one point, the sum of all the angles so 
formed will still be equal to four right angles. This is of course what is 
generally given in the text-boots as Corollary 1. 

Proposition 16, 

In any triangle, if one of the sides be produced, the exterior 
angle is greater than either of the interior and opposite angles. 
Let ABC be a triangle, and let one side of it BC be 
produced to D ; 
l I say that the exterior angle ACD is greater than either 
of the interior and opposite angles 
CBA, BAC. 

Let AC be bisected at E [t 10], 
and let BE be joined and produced 
10 in a straight line to F ; 

let EFbe made equal to BE[t. 3], 
let EC be joined [Post. i],and let AC 
be drawn through to G [Post. a]. 
Then, since AE is equal to EC, 
15 and BE to EF, 

the two sides AE, EB are equal to the two sides CE, 
EF respectively ; 

and the angle AEB is equal to the angle FEC, 

for they are vertical angles. [1. 15] 

» Therefore the base AB is equal to the base FC, 

and the triangle ABE is equal to the triangle CFE, 

and the remaining angles are equal to the remaining angles 

respectively, namely those which the equal sides subtend ; [1. 4] 

therefore the angle BAE is equal to the angle ECF. 




i8o BOOK I [i. 16 

i But the angle BCD is greater than the angle ECF; 

[C.N. 5] 

therefore the angle A CD is greater than the angle BAB. 

Similarly also, if BC be bisected, the angle BCG, that is, 

the angle ACD [i. is], can be proved greater than the angle 

ABC as well. 

Therefore etc. Q. E. D. 

i. the exterior angle, literally "the outside angle," <\ l«rdt yurla. 
i. the interior and opposite angles, rut inrit Kai iritarrior yuri&r. 
11. let AC be drawn through to G. The word is 5iiJx* u p a variation on Ihe more 
usual 4K$ffi\-fiadw. "let it be praduHd. " 
at. CFE, in the text " F£C," 

As is well known, this proposition is not universally true under the 
Riemann hypothesis of a space endless in extent but not infinite in size. On 
this hypothesis a straight line is a "closed series" and returns on itself; and 
two straight lines which have one point of intersection have another point of 
intersection also, which bisects the whole length of the straight line measured 
from the first point on it to the same point again; thus the axiom of Euclidean 
geometry that two straight lines do not enclose a space does not hold. If 4 A 
denotes the finite length of a straight line measured from any point once 
round to the same point again, 2A is the distance between the two intersections 
of two straight lines which meet. Two points A, B do not determine one 
sole straight line unless the distance between them is different from 2 A. In 
order that there may only be one perpendicular from a point C to a straight 
line AS, C must not be one of the two " poles " of the straight line. 

Now, in order that the proof of the present proposition may be universally 
valid, it is necessary that CF should always fall within the angle ACD so that, 
the angle A CF may be less than the angle ACD. But this will not always be 
so on the Riemann hypothesis. For, (1) if BE is equal to A, so that BF is 
equal to 2 A, .Fwill be the second point in which BE and BD intersect ; Le. 
F will lie on CD, and the angle ACF will be equal to the angle ACD. In 
this case the exterior angle ACD will be equal to the interior angle BAC. 
(2) If BE is greater than A and less than 2 A, so that BF is greater than a A 
and less than 4A, the angle ACF will be greater than the angle ACD, and 
therefore the angle A CD will be less than the interior angle BAC. Thus, e.g., 
in the particular case of a right-angled triangle, the angles other than the right 
angle may be (1) both acute, (2) one acute and one obtuse, or (3) both obtuse 
according as the perpendicular sides are (1) both less than A, (2) one less and 
the other greater than A, (3) both greater than A. 

Proclus tells us (p. 307, 1 — 12) that some combined this theorem with the 
next in one enunciation thus : In any triangle, if one side be produced, the 
exterior angle of the triangle is greater than either of the inierior and opposite 
angles, and any two of the interior angles are less than two right angles, the 
combination having been suggested by the similar enunciation of Euclid 1. 32, 
In any triangle, if one of the suits be produced, the exterior angle is equal to the 
two inierior and opposite angles, and the three interior angles of the triangle are 
equal to two right angles. 

The present proposition enables Proclus to prove what he did not succeed 
in establishing conclusively in his note on 1. 12, namely that from one point- 
there cannot be drawn to the same straight line three straight lines equal in length. 



i6, 17I 



PROPOSITIONS 16, 17 



tii 




For, if possible, let AB, AC, AD be all equal, B, C, D being in a 
straight line. 

Then, since AB, AC are equal, the angles 

ABC, ACB are equal. 
Similarly, since AB, AD are equal, the angles 

ABD, ADB are equal. 
Therefore the angle ACB is equal to the angle 

ADC, i.e. the exterior angle to the interior and 
opposite angle : which is impossible. 

Proclus next {p. 308, 14 sqq.) undertakes to prove by means of 1. 16 that, 
if a straight line falling on two straight lines make the exterior angle equal to 
the interior and opposite angle, the two straight lines will not form a triangle or 
meet, for in that case the same angle would be both greater and equal. 

The proof is really equivalent to that of Eucl. 1. 17, If BE falls on the 
two straight lines AB, CD in such a way that the angle 
CDE is equal to the interior and opposite angle ABD, 
AB and CD cannot form a triangle or meet. For, if 
they did, then (by 1. 16) the angle CDE would be 
greater than the angle ABD, while by the hypothesis 
it is at the same time equal to it. 

Hence, says Proclus, in order that BA, DC may 
form a triangle it is necessary for them to approaeh one 
another in the sense of being turned round one pair of 
corresponding extremities, e.g. B, D, so that the other extremities A, C come 
nearer. This may be brought about in one of three ways: (t) AB may 
remain fixed and CD be turned about D so that the angle CDE increases ; 
(2) CD may remain fixed and AB be turned about B so that the angle ABD 
b?r jmes smaller ; (3) both AB and CD may move so as to make the angle 
ABD smaller and the angle CDE larger at the same time. The reason, then, 
of the straight lines AB, CD coming to form a triangle or to meet is (says 
Proclus) the movement of the straight lines. 

Though he does not mention it here, Proclus does in another passage 
(p. 371, 2 — ro, quoted on p. 207 above) hint at the possibility that, while 1. 16 
may remain universally true, either of the straight lines BA, DC (or both 
together) may be turned through any angle not greater than a certain finite 
angle and yet may not meet (the Bolyai-Lobachewsky hypothesis). 



Proposition 17. 

In any triangle two angles taken together in any manner 
are less than two right angles. 
Let ABC be a triangle ; 

I say that two angles of the triangle ABC taken together in 

any manner are less than two right angles. 

For let BC be produced to D. [Post. 2] 

Then, since the angle ACD is an exterior angle of the 

triangle ABC, 

it is greater than the interior and opposite angle ABC. 



282 BOOK I [l 17 

Let the angle ACB be added to each ; 
therefore the angles A CD, ACB are greater than the angles 
ABC, BCA. 







But the angles A CD, ACB are equal to two right angles. 

['; '3] 
Therefore the angles ABC, BCA are less than two right 
angles. 

Similarly we can prove that the angles BAC, ACB are 
also less than two right angles, and so are the angles CAB, 
ABC as well. 
Therefore etc. 

Q. E. D. 

1. taken together in any manner, rirrg pcraXop0artyui'eu, Le. an; pail added 
together- 

As in his note on the previous proposition, Prod us tries to state the cause 
of the property. He takes the case of two straight lines forming right angles 
with a transversal and observes that it is the convergence of the straight lines 
towards one another (<n!™«rn t&v tvBuuf), the lessening of the two right angles, 
which produces the triangle. He will not have it that the fact of the exterior 
angle being greater than the interior and opposite angle is the cause of the 
property, for the odd reason that "it is not necessary that a side should be 
produced, or that there should be any exterior angle constructed. ..and how can 
what is not necessary be the cause of what is necessary ?" (p. 311, 17 — 21). 

Agreeably to this view, Proclus then sets himself to prove the theorem 
without producing a side of the triangle. 

Let ABC be a triangle. Take any point D on 
BC, and join AD. 

Then the exterior angle A DC of the triangle ABD 
is greater than the interior and opposite angle ABD. 

Similarly the exterior angle ADB of the triangle 
ADC is greater than the interior and opposite angle 
A CD. 

Therefore, by addition, the angles ADB, ADC are together greater than 
the angles ABC, ACB. 

But the angles ADB, ADC are equal to two right angles ; therefore the 
angles ABC, ACB are less than two right angles. 

Lastly, Proclus proves (what is obvious from this proposition) that there 
cannot be more than one perpendicular to a straight line from a point without 
it. For, if this were possible, two of such perpendiculars would form a triangle 
in which two angles would be right angles: which is impossible, since any two 
angles of a triangle are together less than two right angles. 





i. 18] PROPOSITIONS 17, 18 J83 

Proposition 18. 
In any triangle the greater side subtends the greater angle. 

For let ABC be a triangle having the side AC greater 
than AB ; 

I say that the angle ABC is also greater than the angle 
BCA. 

For, since AC is greater than AB, let AD be made equal 
to AB [i. 3], and let BD be joined. 

Then, since the angle ADB 
is an exterior angle of the triangle 
BCD, 

it is greater than the interior 
and opposite angle DCB. [1. 16] 

But the angle ADB is equal 
to the angle ABD, 

since the side AB is equal to AD ; 

therefore the angle ABD is also greater than the angle 
ACB\ 

therefore the angle ABC is much greater than the angle 
ACB. 

Therefore etc. 

Q. E. D. 

In the enunciation of this proposition wc have inrarttvay (*" subtend "} used with the 
simple accusative instead of the more usual inti with accusative. The latter construction 
is used in the enunciation of I. ig» which otherwise only diners from that of [• 18 in the order 
of the words. The point to remember in order to distinguish the two is that the datum 
comes first and the fuanititm second, the datum being in this proposition the greater sidt 
and in the next the greater angle. Thns the enunciations are (i. 18) i-a»n6i i-ftydtov j) pclpgr 
J-XrupA tjj* fitlfrtrvL ywflsw uwordvti and (l. 19) rarrbt rpvy&wou vxb rV pd{ot>a -yuvLar Jj 
licifuw xXcupi. iinoTtltti. In order to keep the proper order in English we must use the 
passive of the verb in I. 19. Aristotle quotes the result of 1. to, using the exact wording, 
vk& fkp t)jv ^ttflfw ywrtay bwoTttt/ci {Mtteorologica tit. 5, 376 a 11). 

" In order to assist the student in remembering which of these two 
propositions [1. 18, 19] is demonstrated directly and 
which indirectly, it may be observed that the order is 
similar to that in I. 5 and 1. 6" (Todhunter). 

An alternative proof of L 18 given by Porpnyry 
(see Ptoclus, pp. 315, \\ — 316, 13} is interesting. It 
starts by supposing a length equal to AB cut off from 
the other end of AC; that is, CD and not AD is 
made equal to AB. 

Produce AB to E so that BE is equal to AD, and 
join EC. 

Then, since AB is equal to CD, and BE to AD, 
AE is equal to AC, 





aS* BOOK I [i. 18, 19 

Therefore the angle A EC is equal to the angle ACE. 

Now the angle ABC is greater than the angle A EC, [t. 16] 

and therefore greater than the angle ACE. 
Hence, a fortiori, the angle ABC is greater than the angle ACB. 

Proposition 19. 

In any triangle the greater angle is subtended by the 
greater side. 

Let ABC be a triangle having the angle ABC greater 
than the angle BCA ; 

I say that the side AC is also greater than the side AB. 

For, if not, AC is either equal to AB or less. 

Now AC is not equal to AB ; 
for then the angle ABC would also have been 
equal to the angle A CB ; [u 5] 

but it is not ; 

therefore AC is not equal to AB. 

Neither is AC less than AB, 
for then the angle ABC would also have been less than the 
angle ACB; [t 18] 

but it is not ; 

therefore AC is not less than AB. 

And it was proved that it is not equal either. 
Therefore AC is greater than AB, 

Therefore etc. q. e. d. 

This proposition, like t. 6, can be proved by merely logical deduction from 
1. 5 and 1. 18 taken together, as pointed out by De Morgan. The general 
form of the argument used by De Morgan is given in his Formal Logic (1847), 
p. 25, thus : 

"Hypothesis. Let there be any number of propositions or assertions — 
three for instance, X, Y and Z — of which it is the property that one or the 
other must be true, and one only. Let there be three other propositions 
P, Q and P of which it is also the property that one, and one only, must be 
true. Let it be a connexion of those assertions that : 

when X is true, P is true, 

when Kis true, Q is true, 

when Z is true, R is true. 
Constquenu : then it follows that, 

when P is true, X is true, 

when Q is true, Y is true, 

when JR is true, Z is true" 



[1 


■5] 


[<• 


!8] 


[' 


.6] 


[«■ 


, 9 ] 



i. 19] PROPOSITIONS 18, 19 185 

To apply this to the case before us, let us denote the sides of the triangle 
ABC by a, 6, c, and the angles opposite to these sides by A, B, C respectively, 
and suppose that a is the base. 

Then we have the three propositions, 

when b is equal to c, B is equal to C, 
when b is greater than c, B is greater than C, 1 
when 6 is less than c, B is less than C, f 

and it follows logically that, 

when B is equal to C, b is equal to c, 

when B is greater than C, b is greater than c, \ 

when i? is less than C, b is less than c. J 

Reductio ad absurdum by exhaustion. 

Here, says Proclus (p. 318, 16 — 33), Euclid proves the impossibility "by 
means of division" (« fkaipurtan). This means simply the separation of 
different hypotheses, each of which is inconsistent with the truth of the 
theorem to be proved, and which therefore must be successively shown to be 
impossible. If a straight line is not greater than a straight tine, it must be 
either equal to it or less ; thus in a reductio ad absurdum intended to prove 
such a theorem as 1. 19 it is necessary to dispose successively of two hypotheses 
inconsistent with the truth of the theorem. 

Alternative (direct) proof. 

Proclus gives a direct proof (pp. 319—321) which an-NairizI also has and 
attributes to Heron. It requires a lemma and is consequently open to the 
slight objection of separating a theorem from its converse. But the lemma 
and proof are worth giving. 

Lemma. 

If an angle of a triangle be bisected and the straight line bisecting it meet the 
base and divide it into unequal parts, the sides containing the angle will be 
unequal, and the greater will be that which meets the greater segment of the base, 
and the less that which meets the lest. 

Let AD, the bisector of the angle A of the triangle ABC, meet BC in D, 
making CD greater than BD. 

I say that AC is greater than AB. 
Produce AD to £ so that DE is equal to 
AD. And, since DC is greater than BD, cut 
off DF equal to BD. 

Join BFsmd produce it to G. 
Then, since the two sides AD, DB are 
equal to the two sides ED, DF, and the 
vertical angles at D are equal, 

AB is equal to EF, 
and the angle DEF to the angle BAD, 

i.e. to the angle DAG (by hypothesis). 
Therefore AG is equal to EG, 

and therefore greater than EF, or AB, 
Hence, a fortiori, AC is greater than AB. 





*86 BOOK I U. 19, 30 

Proof of I. 19. 

Let ABC be a triangle in which the angle ABC is greater than the angle 
ACB. * 

Bisect BC at D, join AD, and produce it to B so that DE is equal to 
^Z>. Join BE. 

Then the iwo sides BD t DE are equal to the two 
sides CD, DA, and the vertical angles at D are equal ; 

therefore BE is equal to AC, 

and the angle DBE to the angle at C. 

But the angle at C is less than the angle ABC ; 

therefore the angle DBE is less than the angle 
ABD. 

Hence, if BF bisect the angle ABE, BF meets 
AE between A and D. Therefore EF is greater 
than FA. 

It follows, by the lemma, that BE is greater than 
BA, 

that is, AC is greater than >4.#. 



Proposition 20. 

/« a«y triangle twu sides taken together in any manner 
are greater than the remaining one. 

For let ABC be a triangle ; 
I say that in the triangle ABC two sides taken together in 
any manner are greater than the remaining one, namely 
BA, AC greater than BC, 
AB, BC greater than A C, 
BC, CA greater than AB. 
For let BA be drawn through to the point D s 
let DA be made equal to CA, and let DC be 
joined. 

Then, since DA is equal to AC, 

the angle ADC is also equal to the angle 

ACD; [1. 5 ] 

therefore the angle BCD is greater than 

the angle ADC. [C. JV. 5] 

And, since DCB is a triangle having the angle BCD 
greater than the angle BDC, 

and the greater angle is subtended by the greater side, 

[l 19] 
therefore DB is greater than BC. 




i. 20] PROPOSITIONS 19, 20 287 

ButZM is equal to AC; 

therefore BA, AC are greater than BC. 
Similarly we can prove that AB, BC are also greater 
than CA, and BC, CA than AB. 
Therefore etc. 

Q. E. D, 

It was the habit of the Epicureans, says Prod us (p. 322), to ridicule this 
theorem as being evident even to an ass and requiring no proof, and their 
allegation that the theorem was "known" (yywpijiov) even to an ass was based 
on the fact that, if fodder is placed at one angular point and the ass at another, 
he does not, in order to get to his food, traverse the two sides of the triangle 
but only the one side separating them (an argument which makes Savile exclaim 
that its authors were "digni ipsi, qui cum Asino foenum essent," p. 78). 
Proclus replies truly that a mere perception of the truth of the theorem is a 
different thing from a scientific proof of it and a knowledge of the reason why 
it is true. Moreover, as Sim son says, the number of axioms should not be 
increased without necessity. 

Alternative Proofs. 

Heron and Porphyry, we are told (Proclus, pp. 323 — 6), proved this 
theorem in different ways as follows, without producing one of the sides. 

First proof. 

Let ABC be the triangle, and let it be required to prove that the sides 
BA, AC are greater than BC. 

Bisect the angle BAC by AD meeting BC inD. 

Then, in the triangle ABD, 

the exterior angle ADC is greater than the 
interior and opposite angle BAD, [1. 16] 

that is, greater than the angle DAC. 

Therefore the side AC is greater than the side 
CD, [1. 19] 

Similarly we can prove that AB is greater than BD. 

Hence, by addition, BA, AC are greater than BC. 

Second proof. 

This, like the first proof, is direct. There are several cases to be considered. 

(1) If the triangle is equilateral, the truth of the proposition is obvious. 

(3) If the triangle is isosceles, the proposition needs no proof in the case 
(a) where each of the equal sides is greater than the base. 

(#) If the base is greater than either of the other sides, we have to prove 
that the sum of the two equal sides is greater than 
the base. Let BC be the base in such a triangle. 

Cut off from BC a length BD equal to AB, and 
join AD. 

Then, in the triangle ADB, the exterior angle 
ADC is greater than the interior and opposite angle 
BAD. [1. 1 61 

Similarly, in the triangle ADC, the exterior angle ADB is greater than the 
interior and opposite angle CAD. 





*88 BOOK I [i. to 

By addition, the tyro angles BDA, ADC are together greater than the 
two angles BA D, DA C (or the whole angle BA C). 

Subtracting the equal angles BDA, BAD, we have the angle ADC 
greater than the angle CAD. 

It follows that AC is greater than CD; [i. 19] 

and, adding the equals AB, BD respectively, we have BA, AC together 
greater than BC. 

(3) If the triangle be scalene, we can arrange the sides in order of length. 
Suppose BC is the greatest, AB the intermediate and AC the least side. 
Then it is obvious that AB, BC are together greater than AC, and BC, CA 
together greater than AB. 

It only remains therefore to prove that CA, AB are together greater 
than BC. 

We cut off from BC a length BD equal to the adjacent side, join AD, and 
proceed exactly as in the above case of the isosceles triangle. 

Thirdprwf. 

This proof is by reduetio ad ahsurdum. 

Suppose that BC is the greatest side and, as before, we have to prove that 
BA, AC are greater than BC. 

If they are not, they must be either equal to A 

or less than BC. 

(1) Suppose BA, AC ire together equal 
to BC. 

From BC cut off BD equal to BA, and 
join AD. 

It follows from the hypothesis that DC is equal to AC. 

Then, since BA is equal to BD, 
the angle BDA is equal to the angle BAD. 

Similarly, since AC is equal to CD, 
the angle CDA is equal to the angle CAD. 

By addition, the angles BDA, ADC are together equal to the whole angle 
BAC. 

That is, the angle BAC is equal to two right angles : which is impossible. 

{2) Suppose BA, AC ate together less than BC. 

From BC cut off BD equal to BA, and from CB cut off CE equal to 
CA. Join AD, AE. 

In this case, we prove in the same way that 
the angle BDA is equal to the angle BAD, and 
the angle CEA to the angle CAE. 

By addition, the sum of the angles BDA, 
AEC is equal to the sum of the angles BAD, 
CAE. 

Now, by 1. 16, the angle BDA is greater than the angle DAC, and 
therefore, a fortiori, greater than the angle EA C. 

Similarly the angle AEC is greater than the angle BAD. 

Hence the sum of the angles BDA, AEC is greater than the sum of the 
angles BAD, EAC. 

But the former sum was also equal to the latter : which is impossible, 






li ai] PROPOSITIONS 20, 21 289 

Proposition 21. 

If on one of the sides of a triangle, from its extremities, 

there be constructed two straight lines meeting within the 

triangle, the straight lines so constructed will be less than the 

remaining two sides of the triangle, but will contain a greater 

s angle. 

On BC, one of the sides of the triangle ABC, from its 
extremities B, C, let the two straight lines BD, DC be con- 
structed meeting within the triangle ; 

I say that BD, DC are less than the remaining two sides 
10 of the triangle BA, AC, but contain an angle BDC greater 
than the angle BAC. 

For let BD be drawn through to E. 

Then, since in any triangle two 

sides are greater than the remaining 

15 one, [1. zo] 

therefore, in the triangle ABE, the 

two sides A B, AE are greater than BE. 

Let EC be added to each ; 

therefore BA, AC art greater than BE, EC. 
20 Again, since, in the triangle CED, 

the two sides CE, ED are greater than CD, 
let DB be added to each ; 

therefore CE, EB are greater than CD, DB. 
But BA, AC were proved greater than BE, EC; 
2$ therefore BA, AC are much greater than BD, DC. 

Again, since in any triangle the exterior angle is greater 
than the interior and opposite angle, [1- 16] 

therefore, in the triangle CDE, 

the exterior angle BDC is greater than the angle CED. 
30 For the same reason, moreover, in the triangle ABE also, 
the exterior angle CEB is greater than the angle BAC. 
But the angle BDC was proved greater than the angle CEB ; 
therefore the angle BDC is much greater than the angle 
BAC. 
3 5 Therefore etc. q. e. d, 

1* be con strutted. ..meeting within the triangle. The word n meeting" is not in 
the Greek, where the words are irrht rwmrt&rty. evrLrrwiBtu is the word used of con- 
structing two straight lines to a point (cf- [■ 7) or so as to form a triangle ; but it is necessary 
in English to indicate that they mat. 

3. the straight lines so constructed. Observe the elegant brevity of the Creek al 



29° 



BOOK I 



[i. at 




The editors generally call attention to the fact that the lines drawn within 
the triangle in this proposition must be drawn, 
as the enunciation says, from the ends of the 
side ; otherwise it is not necessary that their 
sum should be less than that of the remaining 
sides of the triangle. Proclus (p. 3*7, II sqq.) 
gives a simple illustration. 

Let ABC be a right-angled triangle. Take 
any point D on BC, join DA, and cut off 
from it DE equal to AB. Bisect AE at F, 
and join FC. 

Then shall CF, FD be together greater than CA, AB. 
For CF, FE are equal to CF, FA, 
and therefore greater than CA. 

Add the equals ED, AB respectively ; 

therefore CF, FD are together greater than CA, AB. 

Pappus gives the same proposition as that just proved, but follows it up 
by a number of others more elaborate in character, selected apparently from 
" the so-called paradoxes " of one Erycinus (Pappus, m. p. 106 sqq.). Thus 
he proves the following : 

1. In any triangle, except an equilateral triangle or an isosceles triangle 
with base less than one of the other sides, it is possible to construct on the 
base and within the triangle two straight lines the sum of which is equal to 
the sum of the other two sides of the triangle. 

2. In any triangle in which it is possible to construct two straight lines on 
the base which are equal to the sum of the other two sides of the triangle it is 
also possible to construct two others the sum of which is greater than that sum. 

3. Under the same conditions, if the base is greater than either of the 
other two sides, two straight tines can be constructed in the manner described 
which are respectively greater than the other two sides of the triangle ; and the 
lines may be constructed so as to be respectively equal to the two sides, if one 
of those two sides is less than the other and each of them less than the base. 

4. The lines may be so constructed that their sum will bear to the sum 
of the two sides of the triangle any ratio less than 2:1. 

As a specimen of the proofs we will give that of the proposition which has 










been numbered (1) for the case where the triangle is isosceles (Pappus, in. 
pp. 108 — 110)1 



I. »i] PROPOSITION 21 391 

Let ABC be an isosceles triangle in which the base AC is greater than 
either of the equal sides .<4.5, BC. 

With centre v4 and radius AB describe a circle meeting j4Cin D. 

Draw any radius AEFsuch that it meets BC in a point F outside the circle. 

Take any point G on EF, and through it draw GZf parallel to AC. Take 
any point .AT on GH, and draw KL parallel to FA meeting AC in L. 

From jSCcut off BN equal to EG. 

Thus AG, or LK, is equal to the sum of AB, BN, and CWis less than LK. 

Now GF, Fffare together greater than GH, 
and CH, UK together greater than CK, 

Therefore, by addition, 
CF, FG, HK are together greater than CK, HG. 

Subtracting HK from each side, we see that 
CF, FG are together greater than CK, KG ; 
therefore, if we add AG to each, 

AF, FCaie together greater than AG, GK, KC. 

And AB, BC are together greater than AF, EC. [1. 31] 

Therefore AB, BC are together greater than A G, GK, KC. 

But, by construction, AB, BN are together equal to AG ; 
therefore, by subtraction, NC is greater than GK, KC, 
and a fortiori greater than KC. 

Take on KC produced a point .A/' such that KM is equal to NC; 
with centre K and radius KM describe a circle meeting CL in 0, and join KO. 

Then shall LK, KO be equal to AB, BC. 

For, by construction, LK is equal to the sum of AB, BN, and KO is 
equal to NC; 

therefore LK, KO are together equal to AB, BC. 

It is after 1. at that (as remarked by De Morgan) the important 
proposition about the perpendicular and obliques drawn from a point to a 
straight line of unlimited length is best introduced : 

Of all straight lines that can be drawn to a given straight line of unlimited 
length from a given point without it : 

(a) the perpendicular is the shortest ; 

(b) of the obliques, that is the greater the fool of which is further from the 
perpendicular ; 

(e) given one oblique, only one other can be found of the same length, namely 
that the foot of which is equally distant with the foot of the given one from the 
perpendicular, but on the other side of it. 

Let A be the given point, BC the given straight line ; let AD be 
the perpendicular from A on BC, 
and AE, AF any two obliques of 
which AF makes the greater angle 
with AD. 

Produce AD to A', making A'D 
equal to AD, and join A'E, A F. 

Then the triangles ADE, A'DE 
are equal in all respects ; and so are 
the triangles ADF, A'DF. 

Now (1) in the triangle AEA' the 
two sides AE, EA' are-greater than AA' [1. 20I, that is, twice AE is greater 
than twice AD. 




a 9 * BOOK I [l. a i, 






Therefore AE is greater than AD. 
(a) Since AE, A'E are drawn to E, a point within the triangle A FA', 
AE, EA' are together greater than AE, EA\ [l *i] 

or twice AE is greater than twice AE.. 
Therefore AE is greater than AE. 

(3) Along DB measure off DG equal to DF, and join AG. 
The triangles AGD, AFD are then equal in all respects, so that the 
angles GAD, FAD are equal, and AG is equal to AF. 

Proposition 22. 

Out of three straight lines, which are equal to three given 
straight lines, to construct a triangle : thus it is necessary that 
two of the straight lines taken together in any manner should 
be greater than the remaining one. [1. *o] 

Let the three given straight lines be A, B, C, and of these 
let two taken together in any manner be greater than the 
remaining one, 

namely A, B greater than C, 

A, C greater than B, 
and B, C greater than A ; 

thus it is required to construct a triangle out of straight lines 
equal to A, B, C. 




a- 

8- 

c- 



Let there be set out a straight line DB, terminated at D 
but of infinite length in the direction of B, 
and let DF be made equal to A, FG equal to B, and GH 
equal to C. [1. 3] 

With centre F and distance FD let the circle DKL be 
described ; 

again, with centre G and distance GH let the circle KLH be 
described ; 
and let KF, KG be joined ; 

I say that the triangle KFG has been constructed out of 
three straight lines, equal to A, B, C. 



i. 11] PROPOSITIONS a i, a* »93 

For, since the point F is the centre of the circle DKS,, 

FD is equal to FK. 

But FD is equal to A ; 

therefore KF is also equal to A. 

Again, since the point G is the centre of the circle LKH, 

GH is equal to GK. 

But GH is equal to C; 

therefore KG is also equal to C. 

And FG is also equal to B ; 

therefore the three straight lines KF, FG, GK are equal to 

the three straight lines A, B, C. 

Therefore out of the three straight lines KF, FG, GK, 

which are equal to the three given straight lines A, B, C, the 

triangle KFG has been constructed. 

6 q. e. F. 

i — *> This is the first cast in the Elements of a Satparfiii to a problem in the sense of a 
statement of the conditions or limits of the possibility of a solution. The criterion is of 
course supplied by the preceding proposition. 

j. thu» It is necessary. This is usually translated (e.g. by Williamson and Simson) 
"But it is necessary," which is however inaccurate, since the Greek is not Sti W but &ri ti). 
The words are the same as those used to introduce the ttopurpit in the other sense of the 
" definition " or " particular statement " of a construction to be effected. Hence, as in the 
latter case we say " thus it is required " (e.g. to bisect the finite straight line AS, I. 10}, we 
should here translate " thus it is necessary. 

4* To this enunciation alt the M5S. and Bocthius add, after the Sioptefifa, the words 
"because in any triangle two sides taken together in any manner are greater than the 
remaining one." But this explanation has the appearance of a gloss, and it is omitted hy 
Proclus and Campanus. Moreover there is no corresponding addition to the Supwitit 
of vi, 18. 

It was early observed that Euclid assumes, without giving any reason, that 
the circles drawn as described will meet if the condition that any two of the 
straight lines A, S, C are together greater than the third be fulfilled. Prod us 
(p. 33 r, S sqq.) argues the matter by means of redudio ad abrurdum, but 
does not exhaust the possible hypotheses inconsistent with the contention. 
He says the circles must do one of three things, (1) cut one another, (1) touch 
one another, (3} stand apart {&«rrarai) from one another. He then considers 
the hypotheses (a) of their touching externally, (t>) of their being separated 
from one another by a space. He should have considered also the hypothesis 
(r) of one circle touching the other internally or lying entirely within the 
other without touching. These three hypotheses being successively disproved, 
it follows that the circles must meet (this is the line taken by Camerer and 
Todhunter). 

Simson says : " Some authors blame Euclid because he does not 
demonstrate that the two circles made use of in the construction of this 
problem must cut one another : but this is very plain from the determination 
he has given, namely, that any two of the straight lines DF, FG, GH must 
be greater than the third, For who is so dull, though only beginning to 
learn the Elements, as not to perceive that the circle described from the 
centre F, at the distance FD, must meet FH betwixt F and H, because FD 
is less than FH; and that, for the like reason, the circle described from the 



ig4 BOOK I [i. 22, 33 

centre G at the distance GH must meet DG betwixt D and G ; and that 
these circles must meet one another, because FD and GH are together 
greater than FG." 

We have in fact only to satisfy ourselves that one of the circles, e.g. that 
with centre G, has at least one point of its circumference outside the other 
circle and also at least one point of its circumference inside the same circle ; 
and this is best shown with reference to the points in which the first circle 
cuts the straight line DE. For (i) FH, being equal to the sum of B and C, 
is greater than A t i.e. than the radius of the circle with centre F, and therefore 
His outside that circle. (2) If GAf be measured along GF equal to GH 
or C, then, since GM is either (a) less or {p} greater than GF, Jfwill fall 
either {a) between G and F or (6) beyond F towards D ; in the first case 
(a) the sum of FM and C is equal to FG and therefore less than the sum 
of A and C, so that FM is less than A or FD ; in the second case {/>) the 
sum of MF and FG, i.e. the sum of MF and B, is equal to GAf or C, and 
therefore less than the sum of ^ and B, so that MF is less than A or FD ; 
hence in either case M falls within the circle with centre F. 

It being now proved that the circumference of the circle with centre G 
has at least one point outside, and at least one point inside, the circle with 
centre F, we have only to invoke the Principle of Continuity, as we have to 
do in 1. 1 (cf. the note on that proposition, p. 242, where the necessary 
postulate is stated in the form suggested by Killing). 

That the construction of the proposition gives only two points of 
intersection between the circles, and therefore only two triangles satisfying 
the condition, one on each side of FG, is clear from I. 7, which, as before 
pointed out, takes the place, in Book 1., of 111. 10 proving that two circles 
cannot intersect in more points than two. 

Proposition 23. 

On a given straight line and at a point on it to construct a 
rectilineal angle equal to a given rectilineal angle. 

Let AB be the "given straight line, A the point on it, and 
the angle DCE the given rectilineal angle; 

thus it is required to construct on the given straight line 
AB, and at the point A on it, a rectilineal angle equal to the 
given rectilineal angle DCE. 





On the straight lines CD, CE respectively let the points 
D, E be taken at random ; 
let DE be joined, 
and out of three straight lines which are equal to the three 



li »3] PROPOSITIONS 22, 23 395 

straight lines CD, DE, CE let the triangle AFG he. con- 
structed in such a way that CD is equal to AF, CE to AG, 
and further DE to FG. [1. 22] 

Then, since the two sides DC, CE are equal to the two 
sides FA., AG respectively, 

and the base DE is equal to the base FG, 

the angle DCE is equal to the angle FAG. [1. 8] 

Therefore on the given straight line A3, and at the point 
A on it, the rectilinealangle FAG has been constructed equal 
to the given rectilineal angle DCE. 

e Q. E. F. 

This problem was, according to Eudemus (see Proclus, p. 333, 5), "rather 
the discovery of Oenopides," from which we must apparently infer, not that 
Oenopides was the first to find any solution of it, but that it was he who dis- 
covered the particular solution given by Euclid. (Cf. Bretschneider, p. 65.) 

The editors do not seem to have noticed the fact that the construction of 
the triangle assumed in this proposition is not exactly the construction given 
in 1. 22. We have here to construct a triangle on a certain finite straight line 
AG as base; in 1. 11 we have only to construct a triangle with sides of given 
length without any restriction as to how it is to be placed. Thus in 1. ■• we 
set out any tine whatever and measure successively three lengths along it 
beginning from the given extremity, and what we must regard as the base is the 
intermediate length, not the length beginning at the given extremity, of the 
straight line arbitrarily set out. Here the base is a given straight line abutting 
at a given point Thus the construction has to be modified somewhat from 







h B 




that of the preceding proposition. We must measure AG along AB so that 
AG is equal to CE (or CD), and GH along GB equal to DE; and then we 
must produce BA, in the opposite direction, to F, so that AF'is equal to CD 
(or CE, if AG has been made equal to CD). 

Then, by drawing circles (1) with centre A and radius AF, (2) with centre 
G and radius GH, we determine K, one of their points of intersection, and we 
prove that the triangle KAG is equal in all respects to the triangle DCE, and 
then that the angle at A is equal to the angle DCE. 

I think that Proclus must (though he does not say so) have felt the same 
difficulty with regard to the use in 1. 33 of the result of 1. 22, and that this is 
probably the reason why he gives over again the construction which I have 
given above, with the remark (p. 334, 6) that "you may obtain the construction 
of the triangle in a more instructive manner (StSatricaA.iKioTi^oi') as follows," 

Proclus objects to the procedure of Apollonius in constructing an angle 
under the same conditions, and certainly, if he quotes Apollonius correctly, the 
tatter's exposition must have been somewhat slipshod. 




296 BOOK I {]. *3, «4 

"He takes an angle CDE at random," says Prod us (p. 335> '9 s Vl-)> ' ,and 
a straight line AB, and with centre D and distance 
CD describes the circumference CE, and in the same 
way with centre A and distance AB the circumference 
FB. Then, cutting off FB equal to CE, he joins AF. 
And he declares that the angles A, D standing on 
equal circumferences are equal." 

In the first place, as Prod us remarks, it should be 
premised that AB is equal to CD in order that the 
circles may be equal; and the use of Book lit. for 
such an elementary construction is objectionable. 
The omission to state that AB must be taken equal 
to CD was no doubt a slip, if it occurred. And, as 
regards the equal angles "standing on equal circum- 
ferences," it would seem possible that Apollonius said 
this in explanation, for the sake of brevity, rather than by way of proof. It 
seems to me probable that his construction was only given from the point of 
view of practical, not theoretical, geometry. It really comes to the same thing 
as Euclid's except that DC is taken equal to DE. For cutting off the arc BF 
equal to the arc CE can only be meant in the sense of measuring the chord 
CE, say, with a pair of compasses, and then drawing a circle with centre B 
and radius equal to the chord CE. Apollonius' direction was therefore 
probably intended as a practical short cut, avoiding the actual drawing of the 
chords CE, BF, which, as well as a proof of the equality in all respects of the 
triangles CDE, BAF, would be required to establish theoretically the correct- 
ness of the construction. 



Proposition 24. 

If two triangles have the two sides equal to two sides 
respectively, but have the one of the angles contained by the equal 
straight lines greater than the other, they will also have the 
base greater than the base. 

5 Let ABC, DEF be two triangles having the two sides 

AB, A C equal to the two sides DE, DF respectively, namely 

AB to DE, and A C to DF, and let the angle at A be greater 

than the angle at D ; 

I say that the base BC is also greater than the base £F, 
10 For, since the angle BAC 

is greater than the angle EDF, 

let there be constructed, on the 

straight line DE, and at the 

point D on it, the angle EDG 
jj equal to the angle BAC; [1. 13] 

let DG be made equal to either 

of the two straight lines AC, 

DF, and let EG, FG be joined. 





i. 24] 



PROPOSITIONS 23, 24 



*97 



Then, since AB is equal to DE, and AC to DG, 
*>the two sides BA, AC are equal to the two sides ED, DG, 
respectively ; 

and the angle BAC is equal to the angle EDG ; 
therefore the base BC is equal to the base EG. [1. 4] 
Again, since DF is equal to DG, 
25 the angle DGF is also equal to the angle DFG ; [l 5] 

therefore the angle DFG is greater than the angle EGF. 
Therefore the angle EFG is much greater than the angle 
EGF. 

And, since EFG is a triangle having the angle EFG 
30 greater than the angle EGF, 

and the greater angle is subtended by the greater side, 

[*. 19] 
the side EG is also greater than EF. 
But EG is equal to BC. 

Therefore BC is also greater than EF. 
3S Therefore etc. 

Q. E. D. 

10. I have naturally left out the well-known words added by Simson in 
order to avoid the necessity of considering three cases : " Of the two sides 
DE, DF let DE be the side which is not greater than the other." I doubt 
whether Euclid could have been induced to insert the words himself, even if 
it had been represented to him that their omission meant leaving two possible 
cases out of consideration. His habit and that of the great Greek geometers 
was, not to set out all possible cases, but to give as a rule one case, generally 
the most difficult, as here, and to leave the others to the reader to work out for 
himself. We have already seen one instance in 1. 7. 

Proclus of course gives the other 
two cases which arise if we do not 
first provide that DE is not greater 
than DF. 

(1) In the first case G may fall 
on EF produced, and it is then 
obvious that EG is greater than EF. 

(2) In the second case EG may 
fall below EF. 

If so, by 1. 21, DF, FE are 
together less than DG, GE. 

But DF is equal to DG ; there- 
fore EF is less than EG, i.e. than 
BC. 

These two cases are therefore 
decidedly simpler than the case taken 
by Euclid as typical, and could well be left to the ingenuity of the learner. 

If however after all we prefer to insert Simson's words and avoid the latter 







BOOK I 



[i-*4 



two cases, the proof is not complete unless we show that, with his assumption, 
/"must, in the figure of the proposition, fall below EG. 

De Morgan would make the following proposition precede: Every straight 
tine drawn from the vertex of a triangle to the base is less than the greater of the 
two sides, or than either if they are equal, and he would prove it by means of 
the proposition relating to perpendicular and obliques given above, p. 291. 

But it is easy to prove directly that F falls below EG, if 
DE is not greater than DG, by the method employed by 
Pfleiderer, Lardner, and Todhunter. 

Let DF, produced if necessary, meet EG in H. 
Then the angle DUG is greater than the angle DEG\ 

["• >6] 
and the angle DEG is not less than the angle DGE ; 

[.. .8] 
therefore the angle DUG is greater than the angle DGH. 
Hence DH is less than DG, [1. 19] 

and therefore DH is less than DF. 

Alternative proof. 

Lastly, the modern alternative proof is worth giving. 

A 






Let DHhisect the angle FDG (after the triangle DEG has been made 
equal in all respects to the triangle A BC t as in the proposition), and let DH 
meet EG in H. Join HF. 

Then, in the triangles FDH, GDH, 

the two sides FD, DH are equal to the two sides GD, DH, 
and the included angles FDH, GDH at equal ; 
therefore the base HF is equal to the base HG 
Accordingly EG is equal to the sum of EH, HF; 

and EH, HF are together greater than EF; [1. 20] 

therefore EG, or BC, is greater than EF. 
Proclus (p. 339, 1 1 sqq.) answers by anticipation the possible question that 
might occur to any one on this proposition, viz. why does Euclid not compare 
the areas of the triangles as he does in 1. 4 ? He observes that inequality of 
the areas does not follow from the inequality of the angles contained by the 
equal sides, and that Euclid leaves out all reference to the question both for 
this reason and because the areas cannot be compared without the help of the 
theory of parallels. " But if," says Proclus, " we must anticipate what is to 
come and make our comparison of the areas at once, we assert that (1) if 
the angles A, D — supposing that our argument proceeds with reference to the 
figure in the proposition — are {together) equal to two right angles, the triangles 



L 34, 25] PROPOSITIONS 14, 25 299 

a« primed equal, {2) if greater than tins right angles, that triangle which has 
the greater angle is less, and (3) </ they are less, greater," Proclus then gives 
the proof, but without any reference to the source from which he quoted 
the proposition. Now an-NairizI adds a similar proposition to 1. 38, but 
definitely attributes it to Heron. I shall accordingly give it in the place 
where Heron put it 



Proposition 25. 

If two triangles have the two sides equal to two sides 
respectively, but have the base greater than the base, they will 
also have the one of the angles contained by the equal straight 
lines greater than the other. 

Let ABC, DEF be two triangles having the two sides 
AB, AC equal to the two sides DE, DF respectively, namely 
AB to DE, and A C to DF ; and let the base BC be greater 
than the base FF; 

I say that the angle BAC is also greater than the angle 
EDF 





For, if not, it is either equal to it or less. 
Now the angle BAC is not equal to the angle EDF; 
for then the base BC would also have been equal to the base 
EF r> 4] 

but it is not ; 

therefore the angle BA C is not equal to the angle EDF. 

Neither again is the angle BA C less than the angle EDF; 

for then the base BC would also have been less than the base 

EF, [ft 24] 

but it is not ; 
therefore the angle BA C is not less than the angle EDF. 
But it was proved that it is not equal either ; 

therefore the angle BAC is greater than the angle EDF. 
Therefore etc. 

Q. E. D. 



3 oo BOOK I [i. 15 

De Morgan points out that this proposition (as also i. 8) is a purely logical 
consequence of i. 4 and 1. 14 in the same way as 1. r 9 and 1. 6 are purely 
logical consequences of 1. 18 and 1. 5. If d, 6, e denote the sides, A, B, C the 
angles opposite to them in a triangle ABC, and a', b', /, A', E, C the sides 
and opposite angles respectively in a triangle A'HC, 1. 4 and I. 14 tell us 
that, i, e being respectively equal to i\ /, 

( 1 ) if A is equal to A\ then a is equal to a', 

(z) if A is less than A', then a is less than «', 

(3) if A is greater than A', then a is greater than a' ; 
and it follows logically that, 

(1) if a is equal to a, the angle A is equal to the angle A', [1. 8] 

(3) if a is less than a, A is less than A', \ 

(3) if a is greater than d, A is greater than A'. } I * a 5J 

Two alternative proofs of this theorem are given by Proclus (pp. 345 — 7), 
and they are both interesting. Moreover both are direct. 

I. Proof by Menelaus of Alexandria. 

Let ABC, DEB" be two triangles having the two sides BA, AC equal to 
the two sides ED, DF, but the base BC greater than the base EF. 




Then shall the angle at A be greater than the angle at D. 
From BC cut off BG equal to EF. At B, on the straight line BC, make 
the angle GBH (on the side of BG remote from A) equal to the angle FED. 
Make BH equal to DE ; join HG, and produce it to meet A C in K. 
Join AH. 

Then, since the two sides GB, BH are equal to the two sides FE, ED 
respectively, 

and the angles contained by them are equal, 
HG is equal to ZVor AC, 
and the angle BHG is equal to the angle EDF. 
Now HK is greater than HG or AC, 

and a fortiori greater than AK; 

therefore the angle KAH\% greater than the angle KHA. 

And, since AB is equal to BH, 

the angle BAH is equal to the angle BHA. 

Therefore, by addition, 

the whole angle BA C is greater than the whole angle BHG, 
that is, greater than the angle EDF. 



i. «5, a6] PROPOSITIONS 35, 36 301 

II, Heron's proof. 

Let the triangles be given as before. 

Since BC is greater than EF, produce EF to G so that EG is equal to 
EC. 

Produce ED to If so that DH is equal to DF. The circle with centre 
D and radius DFwiW then pass through H. Let it be described, as FKH. 







Now, since BA, AC are together greater than £C, 

and Atf, ^C are equal to ,££>, Z>Zf respectively, 

while JC is equal to EG, 

EH is greater than EG. 

Therefore the circle with centre E and radius EG will cut Elf, and 

therefore will cut the circle already drawn. Let it cut that circle in K, and 

join DK, KB. 

Then, since D is the centre of the circle FKH, 

DK is equal to Z>^or AC 
Similarly, since £ is the centre of the circle KG, 
EK is equal to EG or BC, 
And DE is equal to A B. 

Therefore the two sides BA, AC are equal to the two sides ED, DK 
respectively; 

and the base BC is equal to the base EK; 
therefore the angle BAC is equal to the angle EDK 
Therefore the angle BA C is greater than the angle EDF. 

Proposition 26, 

If two triangles have the two angles equal to two angles 
respectively, and one side equal to one side, namely, either the 
side adjoining the equal angles, or that subtending erne 0/ the 
equal angles, they wilt also have the remaining sides equal to 
s the remaining sides and the remaining angle to the remaining 
angle. 



3°* BOOK I [1. 26 

Let ABC, DEF be two triangles having the two angles 
ABC, BCA equal to the two angles DEF, EFD respectively, 
namely the angle ABC to the angle DEF, and the angle 

10 BCA to the angle EFD ; and let them also have one side 
equal to one side, first that adjoining the equal angles, namely 
BC to EF; 

I say that they will also have the remaining sides equal 
to the remaining sides respectively, namely AB to DE and 

>s AC to DF, and the remaining angle to the remaining angle, 
namely the angle BA C to the angle EDF. 





For, if AB is unequal to DE, one of them Is greater. 
Let AB be greater, and let BG be made equal to DE ; 
and let GC be joined. 
20 Then, since BG is equal to DE, and BC to EF, 
the two sides GB, BC are equal to the two sides DE, EF 
respectively; 
and the angle GBC is equal to the angle DEF ; 

therefore the base GC is equal to the base DF, 
25 and the triangle GBC is equal to the triangle DEF, 
and the remaining angles will be equal to the remaining angles, 
namely those which the equal sides subtend ; [1. 4] 

therefore the angle GCB is equal to the angle DFE. 
But the angle DFE is by hypothesis equal to the angle BCA; 
3 o therefore the angle BCG is equal to the angle BCA, 

the less to the greater : which is impossible. 
Therefore AB is not unequal to DE, 
and is therefore equal to it. 
But BC is also equal to EF; 
35 therefore the two sides AB, BC are equal to the two 

sides DE, EF respectively, 
and the angle ABC is equal to the angle DEF; 

therefore the base A C is equal to the base DF, 
and the remaining angle BAC is equal to the remaining 
40 angle EDF. [l 4] 



I. *6] PROPOSITION a* 303 

Again, let sides subtending equal angles be equal, as AB 
to DE; 

I say again that the remaining sides will be equal to the 
remaining sides, namely AC to DF and BC to EF, and 
45 further the remaining angle BAC is equal to the remaining 
angle EDF. 

For, if BC is unequal to EF, one of them is greater. 
Let BC be greater, if possible, and let BH be made equal 
to EF; let AH be joined. 
50 Then, since BH is equal to EF, and AB to DE, 
the two sides AB, BH are equal to the two sides DE, EF 
respectively, and they contain equal angles ; 

therefore the base AH is equal to the base DF, 
and the triangle ABH is equal to the triangle DEF, 
ss and the remaining angles will be equal to the remaining angles, 
namely those which the equal sides subtend ; [1. 4] 

therefore the angle BHA is equal to the angle EFD. 
But the angle EFD is equal to the angle BCA ; 
therefore, in the triangle AHC, the exterior angle BHA is 
60 equal to the interior and opposite angle BCA : 

which is impossible. [1. 16] 

Therefore BC is not unequal to EF, 

and is therefore equal to it. 
But AB is also equal to DE ; 
65 therefore the two sides AB, BC are equal to the two sides 
DE, EF respectively, and they contain equal angles ; 
therefore the base AC is equal to the base DF, 
the triangle ABC equal to the triangle DEF, 
and the remaining angle BAC equal to the remaining angle 
r>EDF, [1.4] 

Therefore etc. 

Q, E. D. 

1 — 3. the aide adjoining the equal angles, rhtupir Hjr rpot roTi but ywrtati. 

39. la by hypothesis equal. vr6Ktir at tnj, according to the elegant Greek idiom. 
ifbtettiat is used for the passive of irrvrlffitfn, as Keit*eu is used for the passive of riBrtfu, and 
to with the other compounds. Cf. Trpxruiitidat, " to be added." 

The alternative method of proving this proposition, viz. by applying one 
triangle to the other, was very early discovered, at least so far as regards the 
case where the equal sides are adjacent to the equal angles in each. An-Nairizi 
gives it for this case, observing that the proof is one which he had found, but 
of which he did not know the author. 




3o 4 BOOK I [i 3 6 

Proclus has the following interesting note {p. 35*, 13 — 18): "Eudemus 
in his geometrical history refers this theorem to Thales. For he says that, in 
the method by which they say that Thales proved the distance of ships in the 
sea, it was necessary to make use this theorem." As, unfortunately, this 
information is not sufficient of itself to enable us to determine how Thales 
solved this problem, there is considerable room for conjecture as to bis 
method. 

The suggestions of Bretschneider and Cantor agree in the assumption 
that the necessary observations were probably made from the top of some 
tower or structure of known height, and that a right-angled triangle was used in 
which the tower was the perpendicular, and the line connecting the bottom of 
the tower and the ship was the base, as in the annexed figure, where AB is the 
tower and C the ship. Bretschneider (Die Geometrie and die Geometer vor 
Eukleides, § 30) says that it was only necessary for 
the observer to observe the angle CAB, and then 
the triangle would be completely determined by 
means of this angle and the known length AB. 
As Bretschneider says that the result would be 
obtained "in a moment " by this method, it is not 
clear in what sense he supposes Thales to have 
"observed" the angle SAC. Cantor is more 
definite (GescA. d. Math. i ( , p. 145), for he says that 
the problem was nearly related to that of finding the 
Seqt from given sides. By the Seqt in the Papyrus Rhind is meant the 
ratio to one another of certain lines in pyramids ot obelisks. Eisenlohr and 
Cantor took the one word to be equivalent, sometimes to the cosine of the 
angle made by the edge of the pyramid with the coterminous diagonal of the 
base, sometimes to the tangent of the angle of slope of the faces of the pyramid. 
Ft is now certain that it meant one thing, viz. the ratio of half the side of 
the base to the height of the pyramid, i.e. the cotangent of the angle of 
slope. The calculation of the Seqt thus implying a sort of theory of simi- 
larity, or even of trigonometry, the suggestion of Cantor is apparently that 
the Seqt in this case would be found from a smalt right-angled triangle ADE 
having a common angle A with ABC as shown in the figure, and that the 
ascertained value of the Stqt with the length AB would determine BC. This 
amounts to the use of the property of similar triangles ; and Bretschneider's 
suggestion must apparently come to the same thing, since, even if Thales 
measured the angle in our sense (e.g. by its ratio to a right angle), he would, 
in the absence of something corresponding to a table of trigonometrical ratios, 
have gained nothing and would have had to work out the proportions all the 
same. 

Max C P, Schmidt also (Kulturhistorische Beiirage zur Kenntnis des 
griechisehcn und romischen Alter turns, 1906, p. 32) similarly supposes Thales to 
have had a right angle made of wood or bronze with the legs graduated, to 
have placed it in the position ADE (A being the position of his eye}, and 
then to have read off the lengths AD, DE respectively, and worked out the 
length of BC by the rule of three. 

How then does the supposed use of similar triangles and their property 
square with Eudemus' remark about 1. 26 ? As it stands, it asserts the 
equality of two triangles which have two angles and one side respectively 
equal, and the theorem can only be brought into relation with the above 
explanations by taking it as asserting that, if two angles and one side of one 
triangle are given, the triangle is completely determined. But, if Thales 




i. 26] PROPOSITION 26 305 

practically used proportions, as supposed, 1. 26 is surely not at all the theorem 
which this procedure would naturally suggest as underlying it and being 
"necessarily used"; the use of proportions or of similar but not equal 
triangles would surely have taken attention altogether away from 1. 26 and 
fixed it on vi. 4. 

For this reason I think Tannery is on the right road when he tries to find 
a solution using 1. 26 as it stands, and withal as primitive as any recorded 
solution of such a problem. His suggestion (La Gtemitrit gretqut, pp. 90—1) 
is based on the fiuminis varatio of the Roman agrimensor Marcus Junius 
Nipsus and is as follows. 

To find the distance from a point A to an inaccessible point B. From A 
measure along a straight line at right angles to AB a 
length AC and bisect it at D. From C draw C£ at right 
angles to CA on the side of it remote from B, and let £ 
be the point on it which is in a straight line with B and D. 

Then, by 1. 26, CE is obviously equal to AB. 

As regards the equality of angles, it is to be observed 
that those at D are equal because they are vertically 
opposite, and, curiously enough, Thales is expressly 
credited with the discovery of the equality of such angles. 

The only objecti